This paper provides explicit formulas for endomorphisms and automorphisms of certain extra-special p-groups, describes their structure, and explores the induced partial order and combinatorial properties related to these groups.
Contribution
It introduces a new representation for the exponent p^2 case, enabling explicit formulas for endomorphisms and automorphisms, and analyzes their algebraic and combinatorial properties.
Findings
01
Explicit formulas for endomorphisms and automorphisms of extra-special p-groups.
02
Description of the endomorphism semigroup and automorphism group structures.
03
Demonstration of the partial order induced by endomorphisms on automorphism orbits.
Abstract
For an odd prime p and a positive integer n, it is well known that there are two types of extra-special p-groups of order p2n+1, first one is the Heisenberg group which has exponent p and the second one is of exponent p2. In this article, a new way of representing the extra-special p-group of exponent p2 is given. These representations facilitate an explicit way of finding formulae for any endomorphism and any automorphism of an extra-special p-group G for both the types. Based on these formulae, the endomorphism semigroup End(G) and the automorphism group Aut(G) are described. The endomorphism semigroup image of any element in G is found and the orbits under the action of the automorphism group Aut(G) are determined. As a consequence it is deduced that, under the notion of degeneration of elements in G, the endomorphism semigroup End(G) induces a…
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TopicsFinite Group Theory Research · Advanced Algebra and Geometry · Coding theory and cryptography
Full text
On the Endomorphism Semigroups of Extra-special p-groups and Automorphism Orbits
C P Anil Kumar
School of Mathematics, Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Prayagraj (Allahabad), 211 019, India. email: [email protected]
and
Soham Swadhin Pradhan
School of Mathematics, Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Prayagraj (Allahabad), 211 019, India. email: [email protected]
Abstract.
For an odd prime p and a positive integer n, it is well known that there are two types of extra-special p-groups of order p2n+1, first one is the Heisenberg group which has exponent p and the second one is of exponent p2. In this article, a new way of representing the extra-special p-group of exponent p2 is given. These representations facilitate an explicit way of finding formulae for any endomorphism and any automorphism of an extra-special p-group G for both the types. Based on these formulae, the endomorphism semigroup End(G) and the automorphism group Aut(G) are described. The endomorphism semigroup image of any element in G is found and the orbits under the action of the automorphism group Aut(G) are determined. As a consequence it is deduced that, under the notion of degeneration of elements in G, the endomorphism semigroup End(G) induces a partial order on the automorphism orbits when G is the Heisenberg group and does not induce when G is the extra-special p-group of exponent p2. Finally we prove that the cardinality of isotropic subspaces of any fixed dimension in a non-degenerate symplectic space is a polynomial in p with non-negative integer coefficients. Using this fact we compute the cardinality of End(G).
In the literature, for a prime p, a special group is defined as an elementary abelian p-group or a p-group where the Frattini subgroup, the commutator subgroup and the center coincide and the center is of exponent p. An extra-specialp-group is a non-abelian special group where the center is of order p. The extra-special p-groups arise in various contexts and are well studied groups.
We mention three contexts. Firstly they occupy a distinctive place in the representation theory (D. E. Gorenstein [7] (Chapter 5, Section 5, Theorem 5.4), L. Dornhoff [5] (Chapter 31, Theorem 31.5), H. Opolka [12]) and the cohomology of finite groups (D. J. Benson and J. F. Carlson [1],[2]). Secondly the extra-special p-groups has generated considerable interest in the study of its non-commuting subsets from a group theoretic and combinatorial view point (A. Y. M. Chin [4], M. Isaacs [3], H. Liu and Y. Wang [10], [11]). Thirdly, the automorphism group of an extra-special p-group is also an important aspect of study in the literature.
D. L. Winter [14] has determined the structure of Aut(G) for an extra-special p-group G. More precisely he has proved that the automorphism group Aut(G) is the semi-direct product of the normal subgroup N of centrally trivial automorphisms, (that is, those automorphisms which act trivially on the center Z(G)) and a cyclic group of order (p−1) generated by an automorphism of G which is an extension of the generator of Aut(Z(G)). Moreover it is shown that the quotient group Inn(G)N of N by the inner automorphism group Inn(G) is isomorphic to a subgroup of a symplectic group whose structure is also known. It is also known that for an odd prime p, the group Aut(G) is a split extension of the outer automorphism group Out(G) by Inn(G). For p=2, this need not be true as shown by R. L. Griess Jr. [8]. H. Liu and Y. Wang [9] have determined the structure of the automorphism group of a generalized extra-special p-group.
In this article, for an odd prime p and a positive integer n, we compute and give an explicit expression for an endomorphism and an automorphism of an extra-special p-group of order p2n+1. More precisely, first we present in an explicitly new way, the extra-special p-group of order p2n+1 and of exponent p2 (Definition 1.2), just similar to one of the standard representations of the Heisenberg group of order p2n+1 (Definition 1.1). These definitions are advantageous to write down formulae for any endomorphism and any automorphism for both the types of groups (in main Theorems Theorem Ω, Theorem Σ). In spite of the already determined structure of the automorphism group in the literature [14], the formulae for endomorphisms and automorphisms given in this article can be derived in a very natural and elegant manner. The importance of these explicit formulae is that they facilitate us to compute the endomorphism semigroup images of elements in the group and the automorphism orbits. These are later used to explore the existence of partial order on automorphism orbits using the notion of degeneration of elements (Definition 1.5). Similar work has been done for the case of finite abelian p-groups by K. Dutta and A. Prasad [6]. We have computed the cardinality of the automorphism group and the cardinality of the endomorphism semigroup of an extra-special p-group for both the types as a polynomial in p with integer coefficients. While computing the cardinality of the endomorphism group we prove that the cardinality of isotropic subspaces of any fixed dimension in a non-degenerate symplectic space is a polynomial in p with non-negative integer coefficients.
1.2. Statement of Main Theorems
We begin this section with a few required definitions in order to state the main theorems.
Definition 1.1** (Extra-special p-group of First Type: Heisenberg Group).**
Let p be an odd prime, n be a positive integer and Fp be the finite field order p.
For u=(u1,u2,…,un)t,w=(w1,w2,…,wn)t∈Fpn, define ⟨u,w⟩=i=1∑nuiwi∈Fp.
Then the Heisenberg group is defined as a set ES1(p,n)=Fpn⊕Fpn⊕Fp
with the following group operation. For (ui,wi,zi)∈ES1(p,n),i=1,2,
[TABLE]
Definition 1.2** (Extra-special p-group of Second Type: Exponent p2).**
Let p be an odd prime, n be a positive integer and Z/piZ be the cyclic ring of order pi,i=1,2. Let
i21:Z/p1Z={0,1,2,…,p−1}↪Z/p2Z={0,1,2,…,p2−1} with i21(a)=pa for a∈Z/p1Z be the standard inclusion as an abelian group where the generator 1∈Z/p1Z maps to p∈Z/p2Z.
For u=(u2,u3,…,un)t,w=(w2,w3,…,wn)t∈(Z/p1Z)n−1, define ⟨u,w⟩=i=2∑nuiwi∈Z/p1Z.
The extra-special group of second type is defined as a set
[TABLE]
with the following group operation.
For (u1i,ui,w1i,wi)∈ES2(p,n),i=1,2,
[TABLE]
Definition 1.3** (Extra-special p-group and its associated symplectic form).**
Let p be an odd prime. A finite group G is said to be an extra-special p-group if [G,G]=G′=Z(G) and Z(G) is of order p. In this case we have that Z(G)G is elementary abelian, isomorphic to (Z/p1Z)2n for some n∈N and is equipped with non-degenerate symplectic form ⟨⟨∗,∗⟩⟩ defined as:
[TABLE]
where f:G×G⟶Fp is defined by the equation [x,y]=zf(x,y) for a fixed generator z of Z(G). Consequentially the group G hence has order p2n+1. If σ is an endomorphism (automorphism) of G then it gives rise to σ an endomorphism (automorphism) of Z(G)G.
Remark 1.4*.*
Let p be an odd prime and G be an extra-special p-group. Then G is isomorphic to either ES1(p,n) or ES2(p,n) for some n.
Definition 1.5** (Partial order on orbits and the notion of degeneration).**
Let G be a finite group. Let Aut(G),End(G) be its automorphism group and endomorphism semigroup respectively. Let S be the set of automorphism orbits in G. Let x,y∈G. We say y is endomorphic to x or x degenerates to y if there exists σ∈End(G) such that σ(x)=y. We say y is automorphic to x if there exists σ∈Aut(G) such that σ(x)=y. We say the endomorphism semigroup induces a partial order ≤ on the automorphism orbits if y is endomorphic to x and x is endomorphic to y then y is automorphic to x. In this case, if O1,O2∈S are two orbits then we write O2≤O1 if for some
y∈O2,x∈O1 we have y is endomorphic to x.
Remark 1.6*.*
Let p be a prime and G be a finite abelian p-group. Then the endomorphism semigroup End(G) (here an endomorphism algebra) induces a partial order on automorphism orbits [6].
Now we introduce some notation before stating the first main theorem.
•
Let ein=(0,…,0,1,0,…,0)t∈Fpn be the vector with 1 in the ith position and [math] elsewhere. Here t stands for transpose.
Let p be an odd prime and n be a positive integer. Let G=ES1(p,n). Then:
(A)
If σ∈End(G) then the induced automorphism σ of Z(G)G satisfies
[TABLE]
where l∈Fp given by the equation σ(z)=zl for any generator z of Z(G).
2. (B)
The explicit expression for σ∈End(G) is given as follows. Consider the elements xi=(ein,0n,0),yi=(0n,ein,0)∈G,1≤i≤n.
Let
[TABLE]
with respect to the ordered basis
{x1,x2,…,xn,y1,y2,…,yn} of Z(G)G=Fp2n. Then for u=(u1,u2,…,un)t,w=(w1,w2,…,wn)t∈Fpn,z∈Fp we have
[TABLE]
where
[TABLE]
for some α,β∈(Fpn)∨ (dual of Fpn) and l∈Fp which satisfies the equation σtΔσ=lΔ. Conversely if σ is given as in Equations 1.1, 1.2, 1.3 then σ∈End(G).
3. (C)
If σ∈Aut(G) then the induced automorphism σ of Z(G)G satisfies
[TABLE]
where l∈Fp∗ given by the equation σ(z)=zl for any generator z of Z(G).
4. (D)
With the notations in (B), the expression for an automorphism σ∈Aut(G) remains the same as in (B) except that, here σ∈Spscalar(2n,Fp) is invertible with l∈Fp∗. Conversely if σ is given as in Equations 1.1, 1.2, 1.3 and l=0 then σ∈Aut(G).
5. (E)
The set of endomorphism semigroup images of an element g∈G is given by:
(a)
{e}* if g=e and has cardinality 1.*
2. (b)
Z(G)* if g∈Z(G)\{e} and has cardinality p.*
3. (c)
G* if g∈G\Z(G) and has cardinality p2n+1.*
6. (F)
There are three automorphism orbits in G. They are given by:
(a)
The identity element {e} and has cardinality 1.
2. (b)
The central non-identity elements Z(G)\{e} and has cardinality p−1.
3. (c)
The non-central elements G\Z(G) and has cardinality p2n+1−p.
7. (G)
The endomorphism semigroup induces a partial order (in fact a total order) on automorphism orbits which is given by
[TABLE]
Now we introduce some further notation before stating the second main theorem.
•
u,w denote vectors in Fpn for some n.
•
Let i21:Z/p1Z↪Z/p2Z be the inclusion of the abelian group Z/p1Z taking the generator 1∈Z/p1Z to p∈Z/p2Z.
•
For u1∈Z/p2Z, let u1∈Z/p1Z be its reduction modulo p.
•
Let π:Z/pZ⊕(Z/p1Z)n−1⟶(Z/p1Z)n−1 be the projection ignoring the first co-ordinate.
•
For G=ES2(p,n) let H=p\big{(}\mathbb{Z}/p^{2}\mathbb{Z}\big{)}\oplus(\mathbb{Z}/p^{1}\mathbb{Z})^{n-1}\oplus\mathbb{Z}/p^{1}\mathbb{Z}\oplus(\mathbb{Z}/p^{1}\mathbb{Z})^{n-1},
K=p\big{(}\mathbb{Z}/p^{2}\mathbb{Z}\big{)}\oplus\{\underline{0}^{n-1}\}\oplus\mathbb{Z}/p^{1}\mathbb{Z}\oplus\{\underline{0}^{n-1}\}=\mathcal{Z}(H) and we have
\mathcal{Z}(G)=p\big{(}\mathbb{Z}/p^{2}\mathbb{Z}\big{)}\oplus\{\underline{0}^{n-1}\}\oplus\{0\}\oplus\{\underline{0}^{n-1}\}.
Now we state the second main theorem of the article.
Theorem Σ.
Let p be an odd prime and n be a positive integer. Let G=ES2(p,n). Then:
(A)
If σ∈End(G) then the induced endomorphism σ of Z(G)G satisfies
[TABLE]
where l∈Fp given by the equation σ(z)=zl for any generator z of Z(G). We also have
(a)
σ(x1)* can be any element of G where x1=(1,0n−1,0,0n−1)∈G.*
2. (b)
For 2≤i≤n,1≤j≤n,σ(xi),σ(yj)∈H where xi=(0,ei−1n−1,0,0n−1), yi=(0,0n−1,0,ei−1n−1).
2. (B)
The explicit expression for σ∈End(G) is given as follows.
Let
[TABLE]
with respect to the ordered basis
{x1,x2,…,xn,y1,y2,…,yn} of Z(G)G=Fp2n.
For (u1,u,w1,w)∈G, let u=(u1u)=(u1,u2,…,un)t∈(Z/p1Z)n,w=(w1w)=(w1,w2,…,wn)t∈(Z/p1Z)n.
Then we have σ may be non-invertible and
[TABLE]
where
[TABLE]
for some α∈((Z/p1Z)n−1)∨,β∈((Z/p1Z)n)∨.
Conversely if σ is given as in Equations 1.4, 1.5, 1.6 then σ∈End(G).
3. (C)
If σ∈Aut(G) then the induced automorphism σ of Z(G)G satisfies
[TABLE]
where l∈Fp∗ given by the equation σ(z)=zl for any generator z of Z(G). We also have
(a)
σ(x1)=x1lg* for some g∈H.*
2. (b)
σ(y1)=y1h* for some h∈Z(G).*
3. (c)
For 2≤i≤n,σ(xi),σ(yi)∈H\K.
4. (D)
With the same notations in (B) the expression for σ∈Aut(G) is given as follows.
Here
[TABLE]
and we have
[TABLE]
where
[TABLE]
for some α∈((Z/p1Z)n−1)∨,β∈((Z/p1Z)n)∨.
Conversely if σ is given as in Equations 1.7, 1.8, 1.9 then σ∈Aut(G).
As a consequence we have in addition
(a)
b11=1.
2. (b)
b21=b31=…=bn1=c21=c31=…=cn1=0.
5. (E)
The set of endomorphism semigroup images of an element g∈G is given by:
(a)
{e}* if g=e and has cardinality 1.*
2. (b)
Z(G)* if g∈Z(G)\{e} and has cardinality p.*
3. (c)
H* if g∈H\Z(G) and has cardinality p2n.*
4. (d)
G* if g∈G\H and has cardinality p2n+1.*
6. (F)
There are (p+2) automorphism orbits if n=1 and (p+3) automorphism orbits if n>1. They are given by:
(a)
The identity element {e} and has cardinality 1.
2. (b)
The central non-identity elements Z(G)\{e} and has cardinality p−1.
3. (c)
For b∈(Z/p1Z)∗,Ob=p(Z/p2Z)×{0n−1}×{b}×{0n−1} and has cardinality p.
4. (d)
G\H, that is, all elements of order p2 and has cardinality p2n+1−p2n.
5. (e)
if n>1 then we have one more orbit H\K and has cardinality p2n−p2.
7. (G)
In this group, there exist two elements which are endomorphic to each other but they are not automorphic. The endomorphism semigroup does not induce a partial order on automorphism orbits.
In particular the set
[TABLE]
is a disjoint union of p automorphism orbits.
2. Preliminaries
It is well known that any extra-special p-group has exponent either p or p2
and has order p2n+1 for some n∈N (refer to D. J. S. Robinson [13], Chapter 5, pp. 140-142). For an odd prime p, if an extra-special p-group of order p2n+1 is of exponent p then it is isomorphic to ES1(p,n) and if it is of exponent p2 then it is isomorphic to ES2(p,n). We also give one more way of presenting the group ESi(p,n) using a symplectic form for i=1,2 which will be useful to prove certain results.
Definition 2.1** (Alternative Definition for ES1(p,n)).**
Let p be an odd prime. Let ES1(p,n)=Fpn⊕Fpn⊕Fp. Let ⟨⟨∗,∗⟩⟩ be a non-degenerate symplectic bilinear form on Fp2n. Then the group structure on ES1(p,n) is defined as: For (ui,wi,zi)∈ES1(p,n),i=1,2 we have
[TABLE]
Definition 2.2** (Alternative Definition for ES2(p,n)).**
Let p be an odd prime, n be a positive integer and Z/piZ be the cyclic ring of order pi,i=1,2. Let
i21:Z/p1Z={0,1,2,…p−1}↪Z/p2Z={0,1,2,…,p2−1} with i21(a)=pa for a∈Z/p1Z be the standard inclusion as an abelian group where the generator 1∈Z/p1Z maps to p∈Z/p2Z. Let
[TABLE]
Then the group structure on ES2(p,n) is defined as follows.
Let ⟨⟨∗,∗⟩⟩ be the non-degenerate symplectic bilinear form on (Z/p1Z)2n given by the matrix J=(0n×n−In×nIn×n0n×n)
with respect to the standard basis.
Let (u1i,ui,w1i,wi)∈ES2(p,n),i=1,2. Let ui=(u1iui),wi=(w1iwi)∈(Z/p1Z)n for i=1,2
where u1i is reduction of u1i modulo p. Then
[TABLE]
Here we state the theorem.
Theorem 2.3**.**
ESl(p,n)≅ESl(p,n),l=1,2.
Proof.
We prove for l=1 first. Let ui=(u1i,u2i,…,uni)t,wi=(w1i,w2i,…,wni)t∈Fpn,i=1,2.
Let u=(u1,u2,…,un)t,w=(w1,w2,…,wn)t∈Fpn. Let ⟨u,w⟩=j=1∑nujwj∈Fp. Let us fix the symplectic form as
[TABLE]
Define a map λ:ES1(p,n)⟶ES1(p,n) given by
[TABLE]
It is easy to check that λ is an isomorphism.
Now we prove for l=2. For i=1,2 let u1i∈Z/p2Z,w1i∈Z/p1Z,ui,wi∈(Z/p1Z)n−1. For i=1,2 let ui=(u1iui)=(u1i,u2i,…,uni)t,wi=(w1iwi)=(w1i,w2i,…,wni)t∈(Z/p1Z)n where u1i is reduction modulo p of u1i∈Z/p2Z. Let u1∈Z/p2Z,w1∈Z/p1Z,u,w∈(Z/p1Z)n−1. Let u=(u1u)=(u1,u2,…,un)t,w=(w1w)=(w1,w2,…,wn)t∈(Z/p1Z)n.
Let ⟨u,w⟩=j=1∑nujwj∈Z/p1Z.
The symplectic form is given as
[TABLE]
Define a map δ:ES2(p,n)⟶ES2(p,n) given by
[TABLE]
It is easy to check that δ is an isomorphism.
This completes the proof of the theorem.
∎
Now we prove a general proposition regarding extra-special p-groups.
Proposition 2.4**.**
Let G be an extra-special p-group. Let z∈Z(G) be a generator such that [g1,g2]=zf(g1,g2) for g1,g2∈G and f:G×G⟶Fp. Let f:Z(G)G×Z(G)G⟶Fp be its associated non-degenerate symplectic bilinear form defined as f(g1,g2)=f(g1,g2). Then we have:
(1)
For σ∈End(G), f(σ(g1),σ(g2))=lf(g1,g2) for any g1,g2∈G where σ(z)=zl,l∈Fp and σ is the induced endomorphism of Z(G)G.
2. (2)
For σ∈Aut(G), f(σ(g1),σ(g2))=lf(g1,g2) for any g1,g2∈G where σ(z)=zl,l∈Fp∗ and σ is the induced automorphism of Z(G)G.
Proof.
We have
[TABLE]
Now the proposition follows.
∎
2.1. Some Commutative Diagrams on Extra-special p-Groups
Now we show that certain diagrams of groups and maps for the extra-special p-group of the first type are commutative.
First we observe that Z(ES1(p,n))={0n}⊕{0n}⊕Fp=Z(ES1(p,n)).
Let
[TABLE]
be the quotient maps of groups.
Let the induced maps be
[TABLE]
Then the following two diagrams commute.
[TABLE]
[TABLE]
[TABLE]
Here λ is as defined in the proof of Theorem 2.3.
In particular we get that Im(Φ1)=Im(Φ1)⊂GL2n(Fp).
Proposition 2.5**.**
Im(Φ1)=Im(Φ1)=Spscalar(2n,Fp).
Proof.
For σ∈Spscalar(2n,Fp) we can define an automorphism σ∈Aut(ES1(p,n)) such that Φ1(σ)=σ as follows.
[TABLE]
Hence we have
Spscalar(2n,Fp)⊆Im(Φ1)=Im(Φ1)⊂GL2n(Fp).
Now use Proposition 2.4 to conclude equality.
∎
Now we show that certain diagrams of groups and maps for the extra-special p-group of the second type are commutative.
First we observe that Z(ES2(p,n))=p(Z/p2Z)⊕{0n−1}⊕{0}⊕{0n−1}=Z(ES2(p,n)).
Let
[TABLE]
be the quotient maps of groups. Let the induced maps be
[TABLE]
Then the following two diagrams commute.
[TABLE]
[TABLE]
[TABLE]
Here δ is as defined in the proof of Theorem 2.3.
In particular we get that Im(Φ2)=Im(Φ2)⊂GL2n(Z/p1Z). We describe this image exactly in Proposition 4.1.
3. Proof of the First Main Theorem
In this section we prove first main Theorem Theorem Ω.
Proof.
Here G=ES1(p,n). Let σ∈End(G) and σ∈End(Z(G)G)=M2n(Fp). Let σ=(ADCB) with A,B,C,D∈Mn(Fp).
Hence we have σ(u,w,z)=(Au+Cw,Du+Bw,σ(u,w,z)) for some σ:G⟶Fp for (u,w,z)∈G.
Using Proposition 2.4 we have
[TABLE]
and AtB−DtC=l.Idn×n where σtΔσ=lΔ. So we also have AtD=DtA,CtB=BtC.
This computation does not give the explicit form of σ as we do not know σ.
Now we compute the explicit form of σ.
The homomorphism condition gives us that, for (ui,wi,zi)∈G,i=1,2,
[TABLE]
Putting w1=w2=0n,z1=z2=0 we get that
[TABLE]
Similarly we have
[TABLE]
We conclude the following.
•
σ(0n,0n,0)=0.
•
Since (u,w,z)=(0n,w,z).(u,0n,0) and (0n,w,z)=(0n,w,0).(0n,0n,z) we have
from Equation 3.1 that
[TABLE]
•
If we define σ1(u)=σ(u,0n,0)−21⟨Au,Du⟩ then from Equation 3.2 and AtD=DtA we conclude that σ1(0n)=0,σ1(u1+u2)=σ1(u1)+σ1(u2). Hence
[TABLE]
•
Similarly from Equation 3.3 and CtB=BtC we conclude that
[TABLE]
•
We observe that
[TABLE]
•
From Equations 3.4, 3.5, 3.6, 3.7 we conclude that
[TABLE]
for some α,β∈(Fpn)∨,l∈Fp.
Conversely if σ=(ADCB)∈sympscalar(2n,Fp) with σtΔσ=lΔ and Equation 3.8 holds, then it is clear that Equation 3.1 holds and σ is an endomorphism of G. This proves (A),(B) in Theorem Theorem Ω.
In case of Aut(G), the proof is similar except that here for σ∈Aut(G), we have l∈Fp∗, that is, it is not allowed to be zero. This proves (C),(D) in Theorem Theorem Ω.
Now we prove (E). In case σ∈End(G) we allow l to be zero. Using Equations 1.2, 1.3, we conclude that the endomorphism semigroup image of g∈G is given by (a) {e} if g=e, (b) Z(G) if g∈Z(G)\{e}, (c) G if g∈G\Z(G).
Now we prove (F). Using Equations 1.2, 1.3 we conclude that there are three automorphism orbits as follows. The identity element {e} is clearly an orbit. The non-identity central elements Z(G)\{e} form an orbit, as automorphisms act transitively on the non-identity central elements because we can choose any non-zero value for l. Now the non-central elements G\Z(G) form an orbit as the group Spscalar(2n,Fp) acts transitively on Fp2n\{02n} and using inner automorphisms we can change the central co-ordinate to any central co-ordinate for the non-central elements.
Now it is clear that endomorphism semigroup End(G) induces a partial order (total order) on the automorphism orbits.
This proves (G) and thereby completes the proof of first main Theorem Theorem Ω.
∎
Using first main Theorem Theorem Ω we have the following corollary.
Corollary 3.1**.**
Let G=ES1(p,n).
(1)
σ∈Aut(G)* is an inner-automorphism if and only if *σ=Id2n×2n.
In this case σ(u,w,z)=α(u)+β(w)+z for some α,β∈(Fpn)∨ for any (u,w,z)∈G.
2. (2)
We have an exact sequence
[TABLE]
3. (3)
[TABLE]
The cardinality of End(G) for G=ES1(p,n) is computed in Section 5, Theorem 5.3.
4. Proof of the Second Main Theorem
In this section we prove second main Theorem Theorem Σ.
Proof.
Here G=ES2(p,n). Let σ∈End(G) and σ∈End(Z(G)G)=M2n(Fp). Let
[TABLE]
Then for x1=(1,0n−1,0,0n−1),σ(x1)=(a11,0n−1,0,0n−1).g for some element g∈H. So for z=(p,0n−1,0,0n−1)∈Z(G) we have σ(z)=(a11p,0n−1,0,0n−1).
Now using Proposition 2.4 we have σ∈sympscalar(2n,Fp) and AtB−DtC=a11.Idn×n where σtΔσ=a11Δ. We also have AtD=DtA,CtB=BtC.
Since the order of x1 is p2 we have o(σ(x1))=p2⟺a11≡0modp. Since the order of xi=(0,ei−1n−1,0,0n−1) is p we have o(σ(xi))∣p⇒a1i≡0modp for 2≤i≤n. Since the order of yi=(0,0n−1,0,ei−1n−1) is p we have o(σ(yi))∣p⇒c1i≡0modp for 2≤i≤n. Similarly for y1=(0,0n−1,1,0n−1)
we have c11≡0modp.
For (u1,u,w1,w)∈G, let u=(u1u)=(u1,u2,…,un)t∈(Z/p1Z)n,w=(w1w)=(w1,w2,…,wn)t∈(Z/p1Z)n. Hence we have
[TABLE]
for some a∈(Z/p2Z),s∈Z/p1Z such that a≡a11modp.
This computation does not give the explicit form of σ as we do not know i21(s).
Just similar to the proof of Theorem Theorem Ω(B) we compute s and obtain
[TABLE]
for some α,β∈((Z/p1Z)n)∨. Now here we can change α(u) to α(u) for some α∈((Z/p1Z)n−1)∨ by shifting multiple of u1 to the first term in au1+i21(s) to obtain
au1+i21(s) without changing the residue class of a modulo p. So we get
[TABLE]
for some a∈(Z/p2Z) such that a≡a11modp
where
[TABLE]
Conversely if σ is as given in Equation 4.1 and s in Equation 4.2 with the matrix σ=(ADCB)∈sympscalar(2n,Fp) satisfying σtΔσ=a11Δ and a12=…=a1n=c11=c12=…=c1n=0 then σ∈End(G). Also in the converse if in addition a11≡0modp, that is, a∈(Z/p2Z)∗ then σ∈Aut(G).
The additional consequences of σ∈Aut(G) are as follows.
We conclude that σ induces automorphisms of the following three subgroups of G.
[TABLE]
Hence σ(y1)=y1b11x1pt with b11=0, for some t∈{0,1,…,p−1} and bj1=0=cj1,2≤j≤n. Now we have AtB−DtC=a11Idn×n⇒a11b11≡a11modp⇒b11=1. This proves (A),(B),(C),(D).
Now we prove (E). Using Equations 1.5, 1.6, the endomorphic images of any element g in G is given as follows. It is {e} if g=e. It is Z(G) if g∈Z(G)\{e}.
It is G if g∈G\H since an element of order p2 can get mapped to any element under an endomorphism.
First we will show that an element g=(a,A21,d11,D21)∈G of order p2 is automorphic to the element (1,0n−1,0,0n−1) where a≡a11≡0modp. Consider the automorphism σ∈Aut(G) such that σ equals
[TABLE]
This automorphism can be used to move (1,0n−1,0,0n−1) to (b,A21,d11,D21) where b≡a≡a11modp. Now we can change (b,A21,d11,D21) to (a,A21,d11,D21) further by another inner automorphism. Now we will show that
[TABLE]
For this the following matrix can be further used.
[TABLE]
It is H if g∈H\Z(G) since a non-central element of order p can get mapped under an endomorphism to any element of order at most p. If g=(pz,u,w1,w)∈H then there are two cases. Either u or w is non-zero or both u or w are zero and w1=0.
Suppose u or w is non-zero. Then we show that g is automorphic to (0,e1n−1,0,0n−1). Let M=(A22D22C22B22)∈Sp(2n−2,Fp) be such that the first column of M is
(uw). Now consider an automorphism σ∈Aut(G) such that σ equals
[TABLE]
where D12=D21tA22−A21tD22, B12=D21tC22−A21tB22.
Here we choose D21 and A21 such that (D12)11=(D21tA22−A21tD22)11=w1.
Note that such choices of D21 and A21 exist because the matrix M is invertible and its first column is non-zero. Now σ moves (0,e1n−1,0,0n−1) to (pz′,u,w1,w)∈H for some z′.
Now using another inner automorphism (pz′,u,w1,w) can be mapped to (pz,u,w1,w)=g.
Now we will show that
[TABLE]
Now let M=(A22D220(n−1)×(n−1)0(n−1)×(n−1))∈sympscalar(2n−2,Fp) where the first column of A22 and D22 are given and rest of the columns of A22,D22 are zero. The following matrix can be further used to show that End(G).(0,e1n−1,0,0n−1)=H.
[TABLE]
Now we consider second case when both u=0=w=0 and w1=0. In this case we show that
[TABLE]
For this following matrix can be used.
[TABLE]
This proves (E).
Now we prove (F). Using Equations 1.5, 1.6, the automorphism orbits in G are given as follows.
The identity element {e} is an orbit. The non-identity central elements Z(G)\{e} is another orbit. For any automorphism σ with σ=(ADCB) we have c11=c21=…=cn1=0,
b11=1,b21=b31=…=bn1=0. So the set Ob=p(Z/p2Z)×{0n−1}×{b}×{0n−1} for b∈(Z/p1Z)∗ is an orbit.
We observe that elements of order p2 forms an orbit, that is, G\H is an orbit and for n>1 the set H\K=H\Z(H) is an orbit. This proves (F).
Now we prove (G). Any element in Ob1 is endomorphic to any element in Ob2 for
b1,b2∈(Z/p1Z)∗. However for 0=b1=b2=0 any element of Ob1 is not automorphic to any element of Ob2. This implies that the endomorphism semigroup does not induce a partial order on the automorphism orbits.
This completes the proof of second main Theorem Theorem Σ.
∎
For Φ2,Φ2 as defined in Section 2.1 we describe the group Im(Φ2)= Im(Φ2)⊂Spscalar(2n,Fp) and set of endomorphisms in End(Z(G)G)=M2n(Fp) which are induced by the elements in the endomorphism semigroup of G=ES2(p,n).
Proposition 4.1**.**
Let G=ES2(p,n). Then
(1)
Im(Φ2)=* Im*\bigg{(}Aut(G)\longrightarrow Aut(\frac{G}{\mathcal{Z}(G)})\bigg{)}=\bigg{\{}\overline{\sigma}=\begin{pmatrix}A&C\\
D&B\end{pmatrix}\in Sp^{scalar}(2n,* \mathbb{F}_{p})\mid A=[a_{ij}],B=[b_{ij}],C=[c_{ij}],D=[d_{ij}]\in M_{n}(\mathbb{F}_{p})\text{ with }a_{11}\neq 0,b_{11}=1,a_{12}=\ldots=a_{1n}=c_{11}=c_{12}=\ldots=c_{1n}=0=c_{21}=c_{31}=\ldots=c_{n1}=b_{21}=b_{31}=\ldots=b_{n1}\text{ and }\overline{\sigma}^{t}\Delta\overline{\sigma}=a_{11}\Delta\bigg{\}}.*
2. (2)
Im\bigg{(}End(G)\longrightarrow End(\frac{G}{\mathcal{Z}(G)})\bigg{)}=* Im*(\Phi_{2})\bigsqcup\bigg{\{}\overline{\sigma}=\begin{pmatrix}A&C\\
D&B\end{pmatrix}\in symp^{scalar}(2n,* \mathbb{F}_{p})\mid A=[a_{ij}],B=[b_{ij}],C=[c_{ij}],D=[d_{ij}]\in M_{n}(\mathbb{F}_{p})\text{ with }a_{11}=a_{12}=\ldots=a_{1n}=c_{11}=c_{12}=\ldots=c_{1n}=0\text{ and }\overline{\sigma}^{t}\Delta\overline{\sigma}=0_{2n\times 2n}\bigg{\}}.*
3. (3)
σ∈Aut(G)* is an inner-automorphism if and only if *σ=Id2n×2n.
In this case for any (u1,u,w1,w)∈G with w=(w1w) we have
[TABLE]
for some α∈((Z/p1Z)n−1)∨,β∈((Z/p1Z)n)∨,a∈(Z/p2Z)∗ such that a≡1modp.
4. (4)
We have an exact sequence
[TABLE]
5. (5)
[TABLE]
6. (6)
[TABLE]
Proof.
This follows from the proof of second main Theorem Theorem Σ.
∎
The cardinality of End(G) for G=ES2(p,n) is computed in Section 5, Theorem 5.3.
5. Order of Endomorphism Semigroups of Extra-Special p-Groups
In this section we compute the cardinality of End(G) for G=ESi(p,n),i=1,2 for an odd prime p and a positive integer n. First we note that Im\bigg{(}End(G)\longrightarrow End(\frac{G}{\mathcal{Z}(G)})\bigg{)} is a disjoint union of Im\bigg{(}Aut(G)\longrightarrow Aut(\frac{G}{\mathcal{Z}(G)})\bigg{)} and an algebraic set defined over Fp given as follows.
Let ⟨⟨∗,∗⟩⟩:Fp2n×Fp2n⟶Fp be the non-degenerate symplectic bilinear form given by
[TABLE]
Let ei=ei2n,fi=en+i2n∈Fp2n,1≤i≤n be the standard basis such that ⟨⟨ei,fj⟩⟩=δij,⟨⟨ei,ej⟩⟩=0=⟨⟨fi,fj⟩⟩,1≤i,j≤n. Let V1=⟨e2,…,en,f1,f2,…,fn⟩.
Let Ei= Im\bigg{(}End(G)\longrightarrow End(\frac{G}{\mathcal{Z}(G)})\bigg{)} where G=ESi(p,n),i=1,2.
Then the following holds.
•
If G=ES1(p,n) then E1= Im(Φ1)⨆X where the algebraic set X={N∈M2n(Fp)∣NtΔN=0} and Φ1 is as defined in Section 2.1. So ∣End(G)∣=p2n∣E1∣ using Equations 1.2,1.3 in Theorem Theorem Ω.
•
If G=ES2(p,n) then E2= Im(Φ2)⨆Y where the algebraic set Y={N∈M2n(Fp)∣NtΔN=0, Im(N)⊆V1} and Φ2 is as defined in Section 2.1. So ∣End(G)∣=p2n∣E2∣ using Equations 1.5,1.6 in Theorem Theorem Σ.
Definition 5.1** (Isotropic Subspace).**
Let ⟨⟨∗,∗⟩⟩:Fp2n×Fp2n⟶Fp be a non-degenerate symplectic bilinear form. A subspace W⊂Fp2n is said to be isotropic if for all v,w∈W,⟨⟨v,w⟩⟩=0.
It is well known that the p-binomial coefficient (kn)p is a polynomial in p with non-negative integer coefficients for any 0≤k≤n and n=0.
Now we state a theorem about enumeration.
Theorem 5.2**.**
Let ⟨⟨∗,∗⟩⟩:Fp2n×Fp2n⟶Fp be the standard non-degenerate symplectic bilinear form. Let ei=ei2n,fi=en+i2n∈Fp2n,1≤i≤n and V1=⟨e2,e3,…,en,f1,f2,…,fn⟩. Let X={N∈M2n(Fp)∣NtΔN=0},Y={N∈M2n(Fp)∣NtΔN=0, Im(N)⊆V1}. For 0≤k≤n,Isotk(Fp2n)={W⊂Fp2n∣W is a \linebreakk-dimensional isotropic subspace} and Isotk(V1)={W⊂V1⊂Fp2n∣W is a \linebreakk-dimensional isotropic subspace}. Let αk(p,n)=∣Isotk(Fp2n)∣,βk(p,n)=∣Isotk(V1)∣. Let γk(p,n)=∣{f:Fp2n↠Fpk∣f is a surjective linear map}∣. Then we have the following.
(1)
∣X∣=k=0∑nαk(p,n)γk(p,n).
2. (2)
∣Y∣=k=0∑nβk(p,n)γk(p,n).
3. (3)
For each 0≤k≤n,αk(p,n),βk(p,n) are polynomials in p with non-negative integer coefficients with
(a)
α0(p,n)=1* and for 1≤k≤n,αk(p,n)=(kn)pi=0∏k−1(pn−i+1).*
2. (b)
For each 0≤k≤n,γk(p,n) is a polynomial in p with integer coefficients with
γ0(p,n)=1 and for 1≤k≤n,γk(p,n)=p2nk−i=0∑k−1(ik)pγi(p,n).
Proof.
If N∈M2n(Fp) and NtΔN=0, that is, Im(N) is an isotropic subspace of Fp2n then dim(Im(N))≤n. So (1) and (2) immediately follow.
Now we prove 3(a). It is clear that α0(p,n)=1.
For 1≤k≤n, let Tk={(v1,v2,…,vk)∈(Fp2n)k∣(v1,v2,…,vk) is an ordered k-tuple of linearly independent vectors whose span is isotropic}. Then we have
[TABLE]
Hence we have
[TABLE]
Now we prove 3(b). It is clear that β0(p,n)=1,β1(p,n)=(12n−1)p. For 2≤k≤n, let Sk={(v1,v2,…,vk)∈(Fp2n)k∣(v1,v2,…,vk) is an ordered k-tuple of linearly independent vectors whose span is isotropic and is contained in V1}.
Let L⊂(Fp2n,⟨⟨∗,∗⟩⟩) be a subspace. We make the following observations.
•
dimL+dimL⊥=2n,(L⊥)⊥=L,V1⊥=⟨f1⟩.
•
f1∈L⟺V1⊥⊆L⟺L⊥⊆V1⟺L⊥∩V1=L⊥.
•
f1∈/L⟺V1⊥⊈L⟺L⊥⊈V1⟺L⊥∩V1⊊L⊥ and of co-dimension one.
Let k=2. We have p2n−1−1 choices for v1∈V1 out of which (p−1) choices of v1 are non-zero multiples of f1 and p2n−1−p choices of v1 are not multiples of f1. In the first case v2∈⟨v1⟩⊥∩V1 has p2n−1−p choices. In the latter case there are p2n−2−p choices for v2∈⟨v1⟩⊥∩V1. So
[TABLE]
So
[TABLE]
Extending the same argument for 3≤k≤n we get
[TABLE]
We also have
[TABLE]
Now we prove (4). It is clear that γ0(p,n)=1. To compute the number of surjective maps we consider all maps from FP2n⟶Fpk and subtract the number of maps of rank less than k.
Hence we get for 1≤k≤n,
[TABLE]
This completes the proof of the theorem.
∎
Theorem 5.3**.**
(1)
For G=ES1(p,n) we have
[TABLE]
2. (2)
For G=ES2(p,n) we have
[TABLE]
Proof.
First we observe that for G=ES1(p,n),∣End(G)∣=∣Aut(G)∣+p2n∣X∣ and for G=ES2(p,n),∣End(G)∣=∣Aut(G)∣+p2n∣Y∣ where X,Y are as defined in Theorem 5.2. Now
using Theorem 5.2, Corollary 3.1(3), we conclude (1) and then again using Theorem 5.2 and Proposition 4.1(6), we conclude (2). This completes the proof of the theorem.
∎
Example 5.4**.**
For n=1 and G=ES1(p,1) we obtain ∣Aut(G)∣=p3(p−1)(p2−1) and ∣End(G)∣=p3(p−1)(p2−1)+p2(1+(11)p(p+1)(p2−1))=p6.
For n=1 and G=ES2(p,1) we obtain ∣Aut(G)∣=p3(p−1) and ∣End(G)∣=p3(p−1)+p2(1+(11)p(p2−1))=2p4−p3.
6. An Open Question
This article leads to an open question which we pose in this section. In general for a finite group,
its center and commutator subgroup are characteristic subgroups. However it is not true that an endomorphism maps the center into itself, but an endomorphism maps commutator subgroup into itself.
Any automorphism or any endomorphism gives rise to a pair of automorphisms and endomorphisms of the commutator subgroup and the abelianization of whole group respectively. The automorphism group and the endomorphism algebra for finite abelian groups are known.
Now we pose the following open question.
Question 6.1**.**
Let p be a prime. Let G be a p-group such that G′=[G,G] is a non-trivial abelian group, that is, G is a non-abelian metabelian p-group. Then:
•
Determine the automorphism orbits in G.
•
Determine the endomorphism semigroup image of any element in G.
•
Determine for which type of such groups G the endomorphism semigroup induces a partial order on the automorphism orbits.
Now in addition for the group G in Question 6.1, if the center coincides with the commutator subgroup then any endomorphism maps the center into itself. Moreover for such a group, if Z(G) is elementary abelian, then we have a non-degenerate skew symmetric bilinear map Z(G)G×Z(G)G⟶Z(G). An example of such a group is given below.
Example 6.2**.**
An example of a non-abelian metabelian p-group G which satisfies [G,G]=G′=Z(G) and Z(G) is elementary abelian is the Heisenberg group Hn(Fq)=Fqn⊕Fqn⊕Fq over the field Fq of order q2n+1 where q=pr for some prime p. The group structure is defined in a similar manner as in ES1(p,n). The answer to Question 6.1 can be explored in the case of Hn(Fq).
Acknowledgements:
The work is done while both the authors are post doctoral fellows at Harish-Chandra Research Institute, Allahabad-INDIA. Both the authors thank Prof. Amritanshu Prasad and Prof. Sunil Kumar Prajapati for mentioning the problem of finding automorphism orbits in extra-special p-groups. The authors also thank Prof. Manoj Kumar Yadav for suggesting a lot of improvements in the article.
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