The Noether numbers and the Davenport constants of the groups of order less than 32
K\'alm\'an Cziszter, M\'aty\'as Domokos, Istv\'an Sz\"oll\H{o}si
TL;DR
This paper completes the computation of Noether numbers for all groups under 32 elements, revealing their properties and relationships with Davenport constants, and introduces algorithms for these calculations.
Contribution
It provides the first complete calculation of Noether numbers for small groups and explores their relation to Davenport constants, answering a longstanding question.
Findings
Noether number is attained on a multiplicity free representation.
Noether number is strictly monotone on subgroups and factor groups.
Identifies a group where Noether number exceeds the large Davenport constant.
Abstract
The computation of the Noether numbers of all groups of order less than thirty-two is completed. It turns out that for these groups in non-modular characteristic the Noether number is attained on a multiplicity free representation, it is strictly monotone on subgroups and factor groups, and it does not depend on the characteristic. Algorithms are developed and used to determine the small and large Davenport constants of these groups. For each of these groups the Noether number is greater than the small Davenport constant, whereas the first example of a group whose Noether number exceeds the large Davenport constant is found, answering partially a question posed by Geroldinger and Grynkiewicz.
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The Noether numbers and the Davenport constants of the groups of order less than 32
Kálmán Cziszter Email: [email protected]
Partially supported by National Research, Development and Innovation Office, NKFIH grants PD113138, ERC HU 15 118286 and K115799. MTA Rényi Institute, 1053 Budapest, Reáltanoda utca 13-15, Hungary
Mátyás Domokos Email: [email protected]
Supported by National Research, Development and Innovation Office, NKFIH K 119934. MTA Rényi Institute, 1053 Budapest, Reáltanoda utca 13-15, Hungary
István Szöllősi Email: [email protected]
Supported by ERC-AdG 321104 and GTC-31816 (Babeş-Bolyai University grant). Faculty of Mathematics and Computer Science, Babeş-Bolyai University, str. M. Kogălniceanu, nr. 1, 400084, Cluj-Napoca, Romania
MTA Rényi Institute, 1053 Budapest, Reáltanoda utca 13-15, Hungary
Abstract
The computation of the Noether numbers of all groups of order less than thirty-two is completed. It turns out that for these groups in non-modular characteristic the Noether number is attained on a multiplicity free representation, it is strictly monotone on subgroups and factor groups, and it does not depend on the characteristic. Algorithms are developed and used to determine the small and large Davenport constants of these groups. For each of these groups the Noether number is greater than the small Davenport constant, whereas the first example of a group whose Noether number exceeds the large Davenport constant is found, answering partially a question posed by Geroldinger and Grynkiewicz.
2010 MSC: 13A50 (Primary) 20D60 (Secondary)
Keywords: polynomial invariant, product-one sequence, degree bound, Noether number, Davenport constant
1 Introduction
1.1 The Noether number
Fix a base field and a finite group . Given a -module (i.e. a finite dimensional -vector space together with an action of via linear transformations) there is an induced action of on the symmetric tensor algebra by -algebra automorphisms. More concretely, can be identified with the polynomial algebra where is a basis of , on which acts via linear substitutions of the variables. Noether [27] proved that the algebra
[TABLE]
of polynomial invariants is generated by finitely many homogeneous elements. Denote by the minimal non-negative integer such that is generated by its homogeneous components of degree at most . Here we refer to the standard grading on , so the variables all have degree one. The Noether number of is
[TABLE]
The following general facts are well known:
[TABLE]
(see [29] for the case , [26] for the case and [14], [15] for the case ).
From now on we assume that is a fixed base field with , and write by suppressing from the notation the dependence of the Noether number on . It is well known that the Noether number is unchanged when we extend the base field (see Subsection 4.3 for more information), so we may assume in proofs that is algebraically closed.
The exact value of the Noether number is known only for a very limited class of groups. First of all, we have
[TABLE]
where is the Davenport constant (the maximal length of an irreducible zero-sum sequence over ); this observation was used first in [30]. The exact value of is known among others for abelian -groups and for abelian groups of rank at most two. Considering non-abelian groups, the Noether number of the dihedral groups was determined in [30] (and in [32] for non-modular positive characteristic) along with the Noether numbers of the quaternion group of order and the alternating group . Recent works of the first two authors of the present paper (see [7], [8], [4], [5], [10]) added a few more (series) of groups to this short list. These results indicated that a complete table of the Noether numbers of “small” groups might be within reach. It turned out that indeed, the reduction lemmas from [7] and considerations similar to the methods used in the above mentioned papers are sufficient to determine the Noether numbers for all groups of order less than . Note that the number of non-abelian groups of order is . This explains our choice of limiting the scope of this paper to the groups of order less than .
1.2 The Davenport constants
Equality (2) inspired Geroldinger and Grynkiewicz [18] to look for an analogue in the case of non-abelian groups. By a sequence over the finite group we mean a finite sequence of elements from which is unordered and repetition of terms is allowed. A sequence is product-one if the product of its elements in an appropriate order is . A sequence is product-one free if it has no product-one subsequences. The small Davenport constant was defined in [28] as the maximal length of a product-one free sequence. A sequence is considered as an element of the free abelian monoid and product-one sequences form a submonoid of . The large Davenport constant was defined in [18] as the maximal length of an atom (irreducible element) in . We have the inequality
[TABLE]
with equaliy for abelian . The question whether is always between and was raised in [18] (the possible relation between the Noether number and Davenport constants is discussed further in [10]). Using the implementation of our algorithms presented in Section 6 we completed the determination of and for groups of order less than . It turned out that holds for these groups with the only exception being the Heisenberg group of order , for which we have .
1.3 Outline of the paper
In Section 2 we give a table containing the values of the Noether number and the Davenport constants for each non-abelian group of order less than . In Section 3 we provide references and proofs to verify the Noether numbers in the table. We draw consequences from the obtained data and state some open questions in Section 4. In particular, in non-modular characteristic for each group of order less than the Noether number is attained on a multiplicity free representation (see Theorem 4.2), the Noether number is strictly monotone with respect to taking subgroups or factor groups (see Theorem 4.5, which is generalized to arbitrary finite groups in the subsequent paper [9]) and the Noether number does not depend on the characteristic (see Theorem 4.7). In Section 5 we turn to the Davenport constants. Notation and known results are recalled in Sections 5.1 and 5.2. In Section 5.3 we present a theoretical proof for the fact that ; this seems to be of special interest because so far this is the only known example of a group for which the Noether number is greater than the large Davenport constant. Section 6 contains the description of the algorithms we employed to compute the Davenport constants given in the table in Section 2.
2 Noether numbers and Davenport constants
The classification of all groups of order less than is given e.g. in [24, Chapter 22]. In the table below we present the Noether numbers and Davenport constants of all the non-abelian groups of order less than . In the first column we also give for reference the GAP (see [16]) identification numbers using which these groups can be constructed in GAP by the function call SmallGroup(m,n).
[TABLE]
In this table and are the symmetric groups of degree and , is the quaternion group of order , is the alternating group of degree and is the binary tetrahedral group, is the Heisenberg group of order (i.e. the group of unitriangular matrices over the -element field). For , is the dihedral group of order , is the dicyclic group given by generators and relations
[TABLE]
Using for the semidirect product of two cyclic groups the notation
[TABLE]
we have that is the semidihedral group
[TABLE]
and for a prime and ,
[TABLE]
Moreover, the symbol always stands for a semidirect product that is not a direct product.
3 Noether numbers
3.1 Abelian groups
It has been long known that for an abelian group we have , see [10] for a recent survey largely motivated by this fact. Therefore we can restate known results on the Davenport constants of abelian groups in terms of the Noether number:
- •
if is cyclic then (see for example [30]);
- •
if is of rank two, i.e. for some then (see for example [19, Theorem 5.8.3]);
- •
if is a -group, i.e. then (see for example [19, Theorem 5.5.9]);
- •
if then by [13].
All abelian groups of order less than fall under one of the four cases above. We note that more recent progress on the Davenport constants of abelian groups can be found in [1], [17, Corollary 4.2.13], [31], [2].
3.2 Groups with a cyclic subgroup of index two
Let be a non-cyclic group having a cyclic subgroup of index two. According to [8, Theorem 10.3] we have
[TABLE]
Formula (3) yields the Noether number for groups out of the groups from the table in Section 2.
3.3 The generalized dihedral groups
The semi-direct product , where is an abelian group on which acts by inversion, is called the generalized dihedral group obtained from . According to [8, Corollary 5.5] we have
[TABLE]
Combining this with the known values of Davenport constants given in Section 3.1 we can compute the Noether number of the three generalized dihedral groups of order less than which are not themselves dihedral groups (these are , , ).
3.4 Cases when the reduction lemmas give exact results
The th Noether number (where is a positive integer) was defined in [7, Section 1.2] as the top degree of the factor space , where stands for the sum of the positive degree homogeneous components of . The supremum of as ranges over all -modules over is denoted by (for an abelian group , equals the th Davenport constant introduced in [22]). In the special case we have and hence . Using the th Noether number one can get upper bounds on by the following reduction lemma:
[TABLE]
Lower bounds on can be derived from the following inequality:
[TABLE]
These results already suffice to establish the precise value of the Noether number for several groups considered below when we combine them with the following formula for the th Noether number:
[TABLE]
Proposition 3.1**.**
.
Proof.
We have the lower bound by (6). On the other hand contains a normal subgroup isomorphic to the Klein four-group such that . Hence by (5) and (7) we have . ∎
Proposition 3.2**.**
.
Proof.
We have the lower bound by (6) since by [30, Lemma 10.1]. On the other hand has a normal subgroup such that . Hence again by (5) and (7). ∎
Proposition 3.3**.**
.
Proof.
We have by (6) and the upper bound by (5) and (7). ∎
Proposition 3.4**.**
.
Proof.
We have by (6) and (3). On the other hand the center of the group is isomorphic to the Klein four-group and we have . So we get by (3), (5) and (7). ∎
Proposition 3.5**.**
.
Proof.
The group has an index two subgroup . By (3) we have , implying by (6) that .
For the reverse inequality observe that the kernel of the action of on is isomorphic to the Klein group . So the subgroup is normal in and as acts by inversion on we have . Therefore by (3), (5) and (7). ∎
Example 3.6**.**
For later reference let us construct here a representation of (where the kernel of the conjugation action of on is isomorphic to ) on which the Noether number is attained. Consider the two-dimensional -module on which the representation is given by the matrices
[TABLE]
where is a primitive third root of unity. Here , and , so these matrices are indeed generating the group in question. Since this group is generated also by the pseudo-reflections and , the -module structure of is well known by the Shephard-Todd-Chevalley theorem [3], [33]. In particular, is a polynomial ring generated by . The element is not contained in the ideal of generated by the above two invariants. It spans a one-dimensional -invariant subspace isomorphic to the one-dimensional -module corresponding to the order two homomorphism given by the determinant on . It follows that in the element is an indecomposable -invariant, where spans the summand .
3.5 Some groups with an abelian normal subgroup of index two or three
In this section we shall discuss a group with an abelian normal subgroup of index or . Let be a -module over an algebraically closed base field . We shall fix the following notation: , . By Clifford theory we know that each irreducible -module is -dimensional or is induced from a -dimensional -module. Therefore it is possible to choose the variables in to be -eigenvectors that are permuted up to scalars by the action of . We shall tacitly assume that the variables were chosen that way. It follows that is spanned by -invariant monomials, permuted by up to scalars. It will be convenient to use the notation for and . For a non-zero scalar multiple of a variable let us denote by its weight (i.e. for all ). Here is the character group of , i.e. the abelian group consisting of the group homomorphisms . Note that when is algebraically closed. For a non-zero scalar multiple of a monomial the multiset is called the weight sequence of . It is a sequence over the abelian group (where the order of the elements in a sequence is disregarded and repetition is allowed). Sequences over form a monoid with multiplication denoted by “” (see Section 5 for more discussion of this monoid), such that for monomials we have . The action of on by conjugation induces an action on by . For a variable , and we have that is a non-zero scalar multiple of a variable, and , showing the equality . Consequently, for a non-zero scalar multiple of a monomial we have where the action of on is extended componentwise to sequences over . Denote by the relative transfer map . It is a surjective -module homomorphism, preserving the grading (inherited from the standard grading on the polynomial ring ). The following lemma provides the common basis for the proofs of the upper bounds for Noether numbers below:
Lemma 3.7**.**
Suppose that for any zero-sum sequence over with which does not factor as the product of non-empty zero-sum sequences, we have a factorization as the product of two zero-sum sequences satisfying the following: for each , the sequence factors as the product of non-empty zero-sum sequences. Then we have the inequality .
Proof.
By Proposition 1.5 in [7], in order to prove it is sufficient to show that where stands for the sum of homogeneous components of of degree at least . That is, we need to show that any -invariant monomial with belongs to . So take an arbitrary -invariant monomial with , and denote by its weight sequence . If factors into a product of zero-sum sequences, then factors as the product of non-trivial -invariant monomials. Thus . On the other hand by Proposition 1.6 in [7] we have , implying in turn that . Next, suppose that does not factor as the product of non-empty zero-sum sequences. Then by assumption we have a factorization with the properties in the statement. The monomial factorizes as with , . We have the equality
[TABLE]
where the summation above ranges over a set of representatives of the -cosets in which are different from . For each such summand, factors as the product of non-empty zero-sum sequences, so . The first summand above is also contained in since , implying in turn that . Thus we showed . ∎
Proposition 3.8**.**
.
Proof.
First we show the inequality . We have where and , so contains an abelian normal subgroup , where centralizes the last summand , and conjugation by gives an order automorphism of . Write the character group additively, and denote its elements by , where is the zero element, is a subgroup, and the action of on induces the action on given by , , . We shall apply Lemma 3.7, so take a zero-sum sequence over with , such that does not factor as the product of four non-empty zero-sum sequences over . Since by Lemma 3.7 in [11], [math] does not occur in (otherwise and implies that is the product of three non-empty zero-sum sequences, a contradiction). On the other hand, implies that contains an element with multiplicity at least , hence , where are zero-sum sequences, has length , so . Note that does not contain a zero-sum subsequence of length (since otherwise would imply that is the product of four non-empty zero-sum sequences). It follows that consists of the non-zero elements in (each having multiplicity one), and so where , . Set and . The factorization fulfills the requirements of Lemma 3.7: indeed, occurs both in and , so can be written as where is a zero-sum sequence of length . Hence by we get that factors as the product of four non-empty zero-sum sequences. Similarly, occurs also in , hence also factors as the product of four non-empty zero-sum sequences. By Lemma 3.7 we conclude that .
The subgroup is normal in and , hence by (6) we obtain the reverse inequality . ∎
Example 3.9**.**
For later use we present a concrete -module on which the Noether number is attained. We keep the notation used in the proof of Proposition 3.8. Let be the -dimensional irreducible -module (the non-trivial direct summand in the -dimensional standard permutation representation of ) viewed as a -module under the natural surjection with kernel . Let be the non-trivial -dimensional -module viewed as a -module under the natural surjection with kernel . Set . Then where are -eigenvectors with weight and they are permuted cyclically by . Similarly are -eigenvectors with weight and they are permuted cyclically by . The last variable is fixed by and is an -eigenvector with weight . Denote by the relative transfer map . Then is spanned as an -vector space by where ranges over the -invariant monomials. There is an -grading on given by
[TABLE]
This is preserved by the action of , hence and are -graded subalgebras. Moreover, preserves the -grading. We claim that is indecomposable, that is, it is not contained in . Suppose to the contrary that is a linear combination of elements of the form where and are non-trivial -invariant monomials and . There is no -invariant variable in , so both and above have total degree at least . The -orbits of the -invariant monomials with , and where is dominated by are
[TABLE]
The -invariants of degree or depending only on are
[TABLE]
It follows that
[TABLE]
for some . Comparing the coefficients of , , , we conclude
[TABLE]
It follows that the coefficient of on the right hand side is [math], whereas on the left hand side it is . This contradiction implies that , hence .
Proposition 3.10**.**
.
Proof.
We have where . The group contains the abelian normal subgroup . We use the same notation for the elements of as in the proof of Proposition 3.8. The action of on is given by
[TABLE]
Take a zero-sum sequence over with which is not the product of three non-empty zero-sum sequences. Since by Lemma 3.7 in [11], has no zero-sum subsequence of length at most . It follows that consists of the non-zero elements of (each element having multiplicity ), hence where and . We have that
[TABLE]
is the product of zero-sum sequences. By Lemma 3.7 we conclude the desired inequality .
On the other hand by (6). ∎
Example 3.11**.**
For later use we present a -module on which the Noether number of the group is attained. Consider the -module on which the action is given by the matrices
[TABLE]
where is a primitive fourth root of unity. Then generate a subgroup of isomorphic to , is the identity matrix, and conjugation by interchanges and . So can be identified with the subgroup of generated by . Moreover, the upper left blocks of give the reflection representation of on . Denote by the standard basis vectors in , so , and . We claim that the -invariant
[TABLE]
in is indecomposable. Consider the characters given by
[TABLE]
and , , the corresponding one-dimensional -modules. Clearly we have the -module isomorphisms
[TABLE]
Since restricts to an order two transformation of , the modules , and their duals do not occur as a summand in , it follows that contains no element that has degree in and degree [math] in , and contains no element that has degree in and degree [math] in . The equalities and show that the elements of having degree both in and are exactly the elements of the form where and . The -module structure of and the structure of is well known from the Shephard-Todd-Chevalley Theorem [3], [33]. We infer that occurs with multiplicity one in the degree component of , namely as the subspace spanned by , and does not occur in lower degrees. This clearly implies that is indecomposable in .
4 Observations and open questions
4.1 The Noether number and multiplicity free representations
When , it was shown in [30] that as a consequence of Weyl’s Theorem on polarizations, is attained on the regular representation of , which contains each irreducible -module with multiplicity equal to its dimension (see [25] for a variant of Weyl’s Theorem [35] valid in positive non-modular characteristic). However for the small groups studied here, the Noether number is usually attained on some -modules of much smaller dimensions. Recall that a -module is multiplicity free if it is the direct sum of pairwise non-isomorphic irreducible -modules.
To deal with some particular cases below, we need first to state explicitly the following corollary of the proof of the inequality (6) given in [8]:
Lemma 4.1**.**
Let be a normal subgroup of with abelian. For any -module there exists a multiplicity free -module such that
[TABLE]
In particular, if , , is multiplicity free and has no summand on which acts trivially, then is attained on a multiplicity free -module.
Theorem 4.2**.**
There exists a multiplicity free -module such that in each of the following cases:
* is abelian;* 2. 2.
* has a cyclic subgroup of index two;* 3. 3.
* has order less than .*
Proof.
It is sufficient to prove our claim in the case when is algebraically closed, so let us assume this.
-
The case when is abelian is known. Denote by the group of characters ( homomorphisms) of , and for let be the -dimensional -module on which acts via . It is well known (see for example Proposition 4.7 in [10]) that if there exists an irreducible zero-sum sequence over of length with components in , then .
-
For the dicyclic groups (where ) we gave an example in the proof of Proposition 9.1 in [8] of a -dimensional irreducible -module with .
Let be a non-abelian group with a cyclic subgroup of index two, and let be a generator of . Then is a -dimensional irreducible -module, and so the module from Lemma 4.1 is multiplicity free (being the direct sum of a -dimensional irreducible and the non-trivial -dimensional module). Moreover, we have , and unless is dicyclic.
- Similar argument works for , , and : let be a maximal cyclic normal subgroup of the given group, and a generator of . Then is an irreducible -module of dimension , and we are done by Lemma 4.1, taking into account the known value of from Section 2.
Now suppose that is a generalized dihedral group with being a non-trivial abelian group. Take a minimal subset such that contains an atom of length , all of whose components belong to . By minimality of , it does not contain the trivial character, and if and both belong to , then . If , then is irreducible and as an -module is isomorphic to , whereas if is non-trivial, then is the direct sum of two non-isomorphic -dimensional -modules, which as -modules are isomorphic to . It follows that for the -module is multiplicity free and contains no summands on which acts trivially. Thus we are done by Lemma 4.1.
The groups , , , can also be settled by Lemma 4.1, taking into account the known value of from Section 2. Indeed, for the -dimensional irreducible -module we have (see [30]) and is the direct sum of the two non-isomorphic irreducible two-dimensional -modules. Essentially the same argument can be used for : after inducing up to the irreducible two-dimensional -module on which the Noether number is attained we get a direct sum of two non-isomorphic irreducible -modules. Note that and for a non-trivial we have that is the direct sum of the three pairwise non-isomorphic irreducible -dimensional -modules. For a generator , we have that is the direct sum of two non-isomorphic irreducible two-dimensional modules.
The cases of the groups , and were settled in Example 3.6, Example 3.9 and Example 3.11.
An irreducible module on which the Noether number is attained is given already in the literature for , in [7], for in [5]. It was pointed out in [10, Example 5.3 and 5.4] that the Noether number for is attained on the product of the standard four-dimensional permutation representation and the sign representation, and for the Pauli group on the direct sum of the two-dimensional pseudo-reflection representation and a one-dimensional representation (see also [6] for some details referred to in [10]). ∎
Problem 4.3**.**
Does there exist a group for which for all multiplicity free -modules ?
Remark 4.4**.**
(i) By a theorem of Draisma, Kemper and Wehlau [12] the universal degree bound for separating invariants is known to be attained on multiplicity free representations.
(ii) We mention a conjecture of Hunziker [23, Conjecture 5.1] made for reflection groups that has a similar flavor as the topic of Section 4.1.
4.2 The strict monotonicity of the Noether number
Since all -modules can be viewed as -modules, the inequality holds for any normal subgroup of any finite group . It was proven by B. Schmid [30] that for any subgroup of . We shall refer as S and F for the following conditions on a finite group :
- S:
for each proper subgroup of .
- F:
for each non-trivial normal subgroup of .
It is shown in a subsequent paper [9] that conditions S and F (by generalizing (6) for non-abelian ) hold for all finite groups. We collect in Theorem 4.5 below facts on the properties S and F that can be read off from the results obtained or quoted in the present paper.
Theorem 4.5**.**
Condition S holds for any finite nilpotent group . Moreover, both S and F hold when
* is abelian;* 2. 2.
* has a cyclic subgroup of index two;* 3. 3.
* for odd primes where ;* 4. 4.
* has order less than .*
Proof.
Suppose first that is nilpotent, and let be a proper subgroup of . It is well known that is contained as a prime index normal subgroup in a subgroup of , whence we have by (6). So S holds for . Note that if is abelian, then any factor group of is isomorphic to a subgroup of , hence F follows from S. If has a cyclic subgroup of index two, then any subgroup or factor group of has a subgroup of index at most two as well, whence (3) shows that both S and F hold. Any non-trivial subgroup or factor group of has order or , and by (6).
Assume finally that is a non-abelian group of order less than that contains no cyclic subgroup of index two. Theorem 1.1 in [7] asserts that unless has a cyclic subgroup of index at most two, or is isomorphic to one of , , , or . Taking into account (3) and the values of the Noether numbers of , , and , this implies that for any non-cyclic , with equality only if is a dicyclic group. Since our has no cyclic subgroups or factor groups of order at least , we conclude that for any proper subgroup or factor group of the inequality holds, with strict inequality unless is a dicyclic group of order or is a cyclic group of order . This immediately implies that S and F hold for provided that . From now on assume in addition that . So is one of the following groups from the table in Section 2: the two groups of order with Noether number , the group of order with Noether number , the two groups of order with Noether number , or the group of order with Noether number . Now has exactly four elements of order and no element of order , consequently does not have as a subgroup or a factor group, hence S and F hold for this group. The group has as a subgroup. Therefore any order subgroup or factor group of this group contains as a subgroup, and hence it is not dicyclic. The group has no element of order and has no dicyclic subgroups or factor groups (as its order is not divisible by ). The groups and have no element of order and do not have a subgroup or factor group isomorphic to (as these groups do not have an element of order ). Finally, the Heisenberg group does not have an element of order . So S and F hold for all groups of order less than . ∎
4.3 Dependence on the characteristic
It is proved in [25, Corollary 4.2] that may depend only on the characteristic of , but not on . Therefore we introduce the notation
[TABLE]
Moreover, by [25, Theorem 4.7] we have for all primes , and holds for all but finitely many primes . Knop remarks in [25] that “Presently, no group and prime not dividing with seems to be known”. This observation inspires the following question:
Problem 4.6**.**
Does the equality hold for all finite groups and primes not dividing ?
The paper [34] reports as a folklore conjecture that for any permutation -module (i.e. when is a free -module with a basis preserved by the action of ) we have provided that does not divide . We note that if this conjecture is true, then Problem 4.6 has a positive answer. Indeed, it follows from [25, Theorem 6.1] and (1) that if does not divide then where is a field of characteristic and is the direct sum of copies of the regular -module defined over (hence in particular is a permutation -module).
In Theorem 4.7 below we collect the cases for which Problem 4.6 has a positive answer by the results obtained or quoted in the present paper.
Theorem 4.7**.**
The equality holds for all primes not dividing the order of in each of the following cases:
* is abelian;* 2. 2.
* has a cyclic subgroup of index two;* 3. 3.
* has order less than .*
Proof.
It has been long known that if is abelian then for all . Formula (3) for the Noether number of a group with a cyclic subgroup of index two is valid in all non-modular characteristic. Finally, the quantities in the table in Section 2 are independent of the characteristic of the base field (provided that it is non-modular), whence the statement holds also for the remaining groups of order less than . ∎
4.4 Further observations
Remark that the groups and both have the structure , the only difference being in the automorphism . However, their Noether numbers are different.
The groups , and the Pauli group all have the structural description , only the automorphism being different in the three cases. The Noether numbers of the first two are equal, and differ from the Noether number of the third one.
This shows that it would be interesting to understand how depends on .
5 Davenport constants
5.1 The monoid of product-one sequences
In this section we introduce further notation related to the small and large Davenport constants of a not necessarily abelian finite group. We follow the presentation of [10].
Let be a non-empty subset of a finite group . A sequence over means a finite sequence of terms from which is unordered, and repetition of terms is allowed (in other words, a sequence over is a multiset of elements from ). A sequence will be considered as an element of the free abelian monoid whose generators are identified with the elements of . We use the symbol “” for the multiplication in the monoid – this agrees with the convention in the monographs [19, 21] – and we denote multiplication in by juxtaposition of elements. For example, considering elements we have that is a sequence of length , while is an element of . Furthermore, we use brackets for the exponentiation in . So for , , and , we have
[TABLE]
Let
[TABLE]
be a sequence over ; here is the multiplicity of in and we call the length of . If , i.e. for a sequence with , then we write
[TABLE]
for the sequence obtained from by removing one occurrence of . More generally, we write if we have for some sequences . The identity element in is called the trivial sequence, and has length . We have the usual divisibility relation in the free abelian monoid and write if divides . A divisor of will also be called a subsequence of . We call the support of . The set of products of is
[TABLE]
(if , we use the convention that ). Clearly, is contained in a -coset, where denotes the commutator subgroup of . Set
[TABLE]
The sequence is called a
- •
product-one sequence if ,
- •
product-one free sequence if .
The set
[TABLE]
of all product-one sequences over is obviously a submonoid of . We denote by the set of atoms in the monoid . The length of an atom is clearly bounded by . The large Davenport constant of is
[TABLE]
Moreover, we denote by the set of product-one free sequences over and we define the small Davenport constant of as
[TABLE]
We have the inequality
[TABLE]
with equality when the elements in commute with each other.
5.2 Some known results
For a non-cyclic group with a cyclic subgroup of index two Olson and White [28] proved that . Morover, recently it was proven by Geroldinger and Grynkiewicz [18] that for these groups .
For the non-abelian semidirect product where are odd primes it was shown by Grynkiewicz [20, Corollary 5.7 and Theorem 5.1] that we have and .
5.3 The large Davenport constant for
Consider , the Heisenberg group with 27 elements having the presentation . This is an extraspecial group, its commutator subgroup coincides with the center . As a result, the commutator identities (which hold for any group) take the following simpler form in this particular case:
[TABLE]
for any . As for any we see that the value of depends only on the cosets and so that the commutator defines in fact a bilinear map on with values in . Moreover as commutes with every other element of the group it is immediate that every -coset has a representative of the form for some and by repeated applications of (9) we get
[TABLE]
By (10) the elements and commute if and only if the vectors and are linearly dependent over . For the rest we denote by the image of any at the natural surjection and we extend this notation to sequences in the obvious way, as well.
We say that two sequences and over a group are similar if for an automorphism (the action of on extends naturally to an action on ). A sequence over is called degenerate if is contained in a proper subgroup of .
Lemma 5.1**.**
Let be a product-one sequence such that and is not product-one free. Then is not an atom.
Proof.
By the assumption on there is a non-empty product-one sequence . Consider the sequence . Then is a zero-sum sequence over , whence . But for we have hence so that is also a product-one sequence. Thus the equality shows that is not an atom. ∎
Lemma 5.2**.**
Let be a non-degenerate sequence over of length at least . Then either or for a basis of
Proof.
Assume that is not a full -coset. As is non-degenerate there must be two elements in such that . Hence . Let be an arbitrary element in . Then because otherwise . Also because otherwise . Similarly is different from , .
As a result in the -vector space we have a relation where the coefficients are non-zero. Moreover also holds by the linear independence of and . Up to similarity and the choice of the basis only two cases are possible: (i) , but then , hence , again a contradiction, or (ii) ; then hence , so that . ∎
In the proof below we shall use the following ad hoc terminology: A subset of of the form where form a basis of will be called an affine line, while a subset of the form where form a basis of will be called an affine cap (the terminology is motivated by the literature on the so-called cap set problem). Note that a three-element subset of in which any two elements are linearly independent over is either an affine line or an affine cap.
Proposition 5.3**.**
.
Proof.
Assume indirectly that there is an atomic product-one sequence of length at least . After ordering the elements of in such a way that and replacing with we may assume that .
A. If is degenerate then it is in fact an irreducible zero-sum sequence over , hence , a contradiction. So for the rest we assume that is non-degenerate so that contains a basis of .
B. contains an element (i.e. has an affine line or an affine cap as a subsequence). Otherwise, if is contained in the set then where and . Choose for . Then the sequence is a zero-sum sequence over , hence if then factors into the product of two non-empty zero-sum sequences and factors accordingly, a contradiction. If then we get , as for all , again a contradiction. Hence and is similar to . Then contains a degenerate subsequence of length , which in turn must contain a proper zero-sum subsequence such that . Then the complement has , hence by Lemma 5.2, a contradiction by Lemma 5.1.
C. cannot contain a subsequence similar to one of the following sequences:
[TABLE]
Indeed, these sequences are such that for their preimages we have that by Lemma 5.2. So for any such the complement must be product-one free, as otherwise we would get a contradiction by Lemma 5.1. Hence cannot be degenerate, as and . Therefore by Lemma 5.2 it is necessary that where and is an affine line. Moreover by our assumption we must have and , as well. But then , a contradiction.
D. We claim that must contain an affine line and . For assume that does not contain an affine line. Then still contains an affine cap by B, so we may assume that . Then to avoid affine lines it is necessary that do not occur in . To avoid subsequences of type (11) we have either or . Assume now that and (the case and is analogous). Then to avoid subsequences of type (12) we must have . As a result the image of modulo is , but this is not a product-one sequence in , a contradiction.
So it remains that . Since is a zero-sum sequence over we must have . As either or and we get again a degenerate subsequence which must contain a zero-sum subsequence such that its complement has by Lemma 5.2, again a contradiction. Thus contains an affine line. An argument similar to the one in C shows also that [math] does not occur in .
From now on we assume that contains the affine line as a subsequence.
E. Next we claim that . Indeed, cannot contain two different elements belonging to because that would yield a subsequence of type (11). Moreover, for some in would yield the type (12) subsequence of .
**F. ** We claim that and . For assume to the contrary that (the case being analogous). Observe that for any in we must have to avoid subsequences of type (12). In view of E it follows that , so that has again a degenerate subsequence containing a zero-sum subsequence (with ) such that its complement has by Lemma 5.2, leading to a contradiction by Lemma 5.1.
G. Now we prove that . For otherwise it would follow from D, E, F that where and . But must also be a zero-sum sequence, hence is divisible by because the only irreducible zero-sum sequences over with support contained in are . But then , contradicting the assumption that .
H. So we have a factorization such that with and . For any such factorization and are zero-sum sequences, hence , moreover by Lemma 5.2, so that and consequently . Observe now that an element in , say must have . Let be such that and . If for some then consider the sequence . By assumption and hence . Thus and would be a factorization of as a product of two product-one sequences. We conclude that , so that is a product-one subsequence and where is a product-one sequence over and hence is a product-one sequence by Lemma 5.2. This contradicts the assumption that is an atom. ∎
Remark 5.4**.**
The idea of Lemma 5.1 appears in the proof of a result of Grynkiewicz [20, Corollary 3.4] which yields for the Heisenberg group of order the inequality
[TABLE]
We established by our algorithm that so in this case (13) gives which in view of Proposition 5.3 is not sharp. On the other hand we show below that Proposition 5.3 is sharp.
Proposition 5.5**.**
.
Proof.
Assume for some . Then since and . Moreover holds, as well. But since and are independent in it is necessary that . It follows that . On the other hand hence the minimal expression of in terms of and has length . By a similar argument the same is true for , as well.
Now and we claim that the product-one sequence is an atom. Otherwise let for some non-empty product-one sequences and we may assume that . If holds, too, then contains only and . Hence, by what has been said before, . But then , which is not a product-one sequence. From this contradiction it follows that hence . But then and , hence again , so that is empty, which is a contradiction. ∎
5.4 Diameter of the Cayley digraph
Next we state a general inequality that can be obtained by a similar argument as the proof of Proposition 5.5. This result is not used here, but seems worthwhile to mention, as it involves the diameter of Cayley digraphs, which (in contrast with the large Davenport constant for a non-abelian group) has a rather extensive literature:
Proposition 5.6**.**
Let be a set of generators of a finite group and the corresponding Cayley digraph. Then .
Proof.
Let be a sequence of vertices on a non-self-crossing path of maximal length in , so that . Now set for all and consider the sequence . By construction is a product-one sequence and we claim that it is an atom in . For otherwise for some product-one sequences where , and then we can order the elements of in such a way that we obtain an equality where . But since for all , this means that we have in a path of length from to , a contradiction. ∎
Problem 5.7**.**
Is there any relation between and ? More concretely, based on a little computer experimentation we raise the following question: does the inequality hold in general?
6 Computing the Davenport constants
We take over the following notations from Section 5.1: let be the monoid of all sequences of elements in , let be the set of product-one free sequences and the set of atoms, i.e. all product-one sequences which cannot be written as the product of two non-empty product-one subsequences. For any let denote the set of all sequences of length and set , . Moreover, consists of the empty sequence.
We propose two algorithms: one for enumerating all product-one free sequences and another one for computing all the atoms and thereby establishing the values of and , respectively. We have implemented these algorithms in the GAP computer algebra system (see [16]) and used them to compute and/or verify the small and large Davenport constants given in Section 2.
The algorithms construct the set (respectively ) successively, as a union of the sets (respectively ). During the construction of the sets or we are not testing all elements of for being product-one free or atomic (as this task would be practically unfeasible), but we limit the scope of our search to a much smaller subset defined in what follows.
A sequence is called a splitting of a sequence if for some and such that . We denote by the fact that is a splitting of . For any subset we set .
Lemma 6.1**.**
For any we have and .
Proof.
If is a product-one free (respectively atomic) sequence of length , then obviously the sequence has length and is product-one free (respectively atomic) too. Moreover, . ∎
Remark 6.2**.**
Observe that by this lemma if for some then for every . So the smallest such that equals . Similarly the smallest such that is equal to .
Another idea used for reducing the complexity of our algorithms was to enumerate the elements of and only “up to similarity”. Recall from Section 5.3 that two sequences and are said to be similar if there is an automorphism such that .
Lemma 6.3**.**
Let be a set of -orbit representatives over (respectively ). Then there is a system of -orbit representatives over (respectively ) such that .
Proof.
Observe that the relation is compatible with the -action in the sense that holds if and only if for some if and only if for each .
Now we prove that for any sequence its -orbit has a non-empty intersection with . Take an arbitrary sequence so that . As is a complete set of representatives, there is some such that for some . Hence by the compatibility of it follows that . As a result and we are done. The proof is similar for the case of . ∎
6.1 Algorithm for computing the small Davenport constant
Let us record the following obvious statement:
Lemma 6.4**.**
A sequence of length is product-one free if and only if and for all we have that .
Given an arbitrary finite group , Algorithm 1 computes the small Davenport constant. The set of all product-one free sequences of may be obtained as after the repeat-until loop has terminated (right after line 1).
In lines 1–1 of the algorithm we have identified the sequences of length one with the respective group elements. Hence the set of product-one free sequences having length one consists of all the group elements except the identity (line 1), from which we choose a set of -orbit representatives (line 1). For every length-one sequence we store the set of products (line 1). These initial data having been established, the algorithm proceeds with computing the sets and for all increasing , until it finds the first value for which . The finiteness of the algorithm is guaranteed by Lemma 6.1 (together with Remark 6.2).
Observe that inside the for-loops in line 1 the variable will take successively all the values from the set , therefore it is guaranteed that for all the possible candidate sequences will be tested for being product-one free and the set (from Lemma 6.3) is built correctly.
Before testing the sequence for being product-one free (using Lemma 6.4), in line 1 it is checked first whether the sequence is among the already computed product-one free sequences of length (via the condition ). Elimination of duplicates is essential for correctly building the set of representatives (so that in line 1 we compute only the set and not one of its supersets). If is a new sequence (in the sense that it is not similar to any of the already found product-one free sequences), it gets tested using the conditions from Lemma 6.4 (the second condition from line 1 and the condition from line 1). is computed (in line 1) using the following observation: for any sequence and fixed , we can write
[TABLE]
If is found to be product-one free, it is also added as a new representative of the -orbits (line 1). The orbit itself is computed successively (and stored in the variable ) by evaluating all the automorphisms on the sequence . In line 1 the set will be the same as the set of orbits of .
For every new sequence (new in the sense of not being in the already computed part of the orbit), we also have to compute and store the set by acting with the proper automorphism on the set (line 1). Hence the algorithm stores the product sets for all the enumerated product-one free sequences. This is essential for a relatively fast computation performed in line 1.
The set gets computed in line 1 as a union of orbits of the representatives. In the last step, the value is returned: this is the small Davenport constant of , as explained in Remark 6.2.
6.2 Algorithm for computing the large Davenport constant
Given an arbitrary finite group , Algorithm 2 computes the large Davenport constant. The set of all atoms of may be obtained as after the repeat-until loop has terminated (right after line 2).
The algorithm begins with loading the sets and with the sequence consisting only of the identity element of the group. In line 2 the variable will take successively all the values from the set , therefore it is guaranteed that for all the possible candidate sequences will be tested for being atoms and the set (from Lemma 6.3) is built correctly. In line 2 the variable is prevented from taking the values or , since this would bring in the identity element in the support of .
The algorithm first checks whether a candidate sequence is among the already computed atoms of length via the condition from line 2, ensuring that is a new atom (in the sense that it is not similar to any of the already found atoms of length ). Checking whether a sequence is an atom is performed by the second condition in line 2: by definition the product-one sequence is an atom if it cannot be written as a product of shorter atoms. The predicate in line 2 is true if and only if the product-one sequence of length is not an atom and may be given in the following way:
[TABLE]
It may be implemented using a recursive function, bailing out with a positive answer as soon as it finds a decomposition of into atoms.
In line 2 the newly found atom gets added to the set of representatives and the set of atoms is completed with the orbit of , denoted by (in line 2). In the last step, the value is returned as the large Davenport constant of , as explained in Remark 6.2.
Although the check is performed only for a system of representatives of the orbits of atoms (for all ), the “decomposability test” described by the predicate uses the whole set of already computed atoms , which is the main performance bottleneck of the algorithm.
Remark 6.5**.**
We have used a GAP implementation of Algorithm 1 and 2 to compute the small and large Davenport constants from the table in Section 2 in case of those groups for which a formula for these constants in not readily available (see Subsection 5.2).
Although parallelization is easily achievable in case of both algorithms, a “single-threaded” implementation was found to fit our present requirements, using a personal computer clocked at 2 GHz.
The small Davenport constants from Section 2 have been obtained in under a minute worth of computation, with the single exception of the group , for which has been obtained in about 7 minutes. This group was found to have a number of product-one free sequences, grouped in equivalence classes. In contrast, the second most “difficult” group (among the non-abelian groups of order less than that contain no cyclic subgroup of index two) was for which our implementation of Algorithm 1 yielded the following data in about 40 seconds: with product-one free sequences partitioned by the similarity relation into equivalence classes. In case of the other groups the running times range from below one second up to half a minute.
Computation of the large Davenport constant using Algorithm 2 is more demanding. The most time-consuming group from the list was , with the large Davenport constant being 13, housing atoms partitioned in equivalence classes. In the case of this group the computation took about minutes. Excepting and (computation taking about 24 minutes for each of them), the running times hardly reached one minute. For example in the interesting case of , the large Davenport constant was computed in seconds. This group has atoms grouped in only equivalence classes (this explaining the relative quick computation within this group).
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