On equality of inner and absolute central automorphisms
Z. Kaboutari Farimani, M. M. Nasrabadi

TL;DR
This paper investigates conditions under which the group of absolute central automorphisms of a finite p-group coincides with its inner automorphism group, providing a precise characterization.
Contribution
It establishes necessary and sufficient conditions for the equality of Aut_l(G) and Inn(G) in finite p-groups, advancing understanding of automorphism structures.
Findings
Characterization of when Aut_l(G) equals Inn(G)
Necessary and sufficient conditions identified
Enhanced understanding of automorphism groups in p-groups
Abstract
Let G be a finite p-group and let Aut_l(G) be the group of absolute central automorphisms of G. We give necessary and sufficient conditions on G such that Aut_l(G) = Inn(G).
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Taxonomy
TopicsAdvanced Topics in Algebra · Advanced Differential Equations and Dynamical Systems · Finite Group Theory Research
On equality of inner and absolute central automorphisms
Z. Kaboutari Farimani
M. M. Nasrabadi *∗*00footnotetext: ∗ corresponding author.
Department of Mathematics, University of Birjand, Birjand, Iran.
z[email protected], [email protected], [email protected]
Abstract.
Let be a finite -group and let be the group of absolute central automorphisms of . We give necessary and sufficient conditions on such that .
Key words and phrases:
absolute central automorphisms; inner automorphisms; finite -groups
2010 Mathematics Subject Classification: 20D45; 20D15
1. **Introduction **
An automorphism of is called central if for each . The central automorphisms of , denoted by , fix elementwise and form a normal subgroup of the full automorphism group of . The properties of have been well studied. See [1, 2, 4] for example.
Hegarty in [3] generalized the concept of centre into absolute centre. The absolute centre of a group , denoted by , is the subgroup consisting of all those elements that are fixed under all automorphisms of . Also he introduced the absolute central automorphisms. An automorphism of is called an absolute central automorphism if for each . We denote the set of all absolute central automorphisms of by . Notice that is a normal subgroup of contained in .
In [2] authors gave necessary and sufficient conditions on a -group such that . In [6] we gave conditions on a finite autonilpotent -group of class such that . In this paper we intend to give necessary and sufficient conditions on a non-abelian -group in which and coincide.
Throughout this paper all groups are assumed to be finite and denotes a prime number. Also if is a group, then , and stand for the exponent, the group of homomorphisms of into an abelian group and the subgroup generated by all powers of elements of , respectively. Let and be two normal subgroups of . We denote the subgroup of consisting of all automorpisms centralizing by and the subgroup of consisting of all automorphisms which act trivially on by . Also, we consider .
2. Preliminary results
In this section, we give some results that will be used in the proof of the main results.
Lemma 2.1**.**
[6, Lemma 2.3]** Suppose is an abelian -group of exponent , and is a cyclic group of order divisible by . Then is isomorphic to .
Let be a group. Then the autocommutator of an element and automorphism is defined as .
Definition 2.2**.**
The absolute centre of a group , denoted by , is defined as
[TABLE]
Clearly, is a central characteristic subgroup of .
Definition 2.3**.**
An automorphism of is called absolute central, if for each . The set of all absolute central automorphisms of is denoted by .
Clearly, is an abelian normal subgroup of .
Proposition 2.4**.**
[5, Proposition 1]** Let be a group. Then
[TABLE]
Lemma 2.5**.**
Let be a group. Then
[TABLE]
Proof.
Consider the map defined by for all and each . Clearly, is a well-defined homomorphism of into . Now, a simple verification shows that the map defined by , for any , is an isomorphism. ∎
Lemma 2.6**.**
Let be a -group of class such that . Then
[TABLE]
where and .
Proof.
Let and . Clearly, . Now let
[TABLE]
and
[TABLE]
where and . Since is a -group and , divides . The rest of proof is similar to that of [6, Lemma 2.8] so we omit the details. ∎
Now we establish a lower bound for .
Lemma 2.7**.**
Let be a -group of class such that . Then , where and are as defined before.
Proof.
We have
[TABLE]
So by Lemma , . ∎
Lemma 2.8**.**
[6, Lemma 2.7]** Let be a group. Then is abelian if and only if .
3. Main results
In this section, we obtain some properties of the group when , and then give necessary and sufficient conditions under which .
Lemma 3.1**.**
Let be a non-abelian -group. If , then is cyclic.
Proof.
Let . Thus . Hence . Also, is abelian so that is nilpotent of class . Now, by Lemma , we have , where and . Thus . Since we must have , that is, is cyclic. ∎
Proposition 3.2**.**
Let be a non-abelian -group ( odd) such that . Then .
Proof.
Suppose is an odd prime number and . Consider the map defined by for all . Clearly, is an automorphism when . In this case, for all in . Hence , which is a contradiction. Thus . ∎
Theorem 3.3**.**
Let be a non-abelian -group. Then if and only if and is cyclic.
Proof.
Suppose and is cyclic. Since , is nilpotent of class and . Hence divides and by Lemma , . Therefore by Lemma , . On the other hand, by Lemma , we have . Now since fixes the centre element-wise, we conclude that . Hence .
To prove the converse, assume that . Let and be the inner automorphism of induced by . Then, for all in , and consequently . Hence,
[TABLE]
where and . Thus . Since , we must have so that is cyclic. ∎
Theorem 3.4**.**
Let be a non-abelian -group. Then if and only if , is cyclic and , where .
Proof.
Suppose first that , is cyclic and . By Theorem , . Now suppose . Let and be such that . Then . Since , we have and so . Hence acts trivially on , that is, . Thus so that , as required.
Conversely, suppose . By Lemma , is cyclic and by Lemma , . Now we have . Hence for all , . This means , for all . Therefore whence . Since , it follows that
[TABLE]
As
[TABLE]
and
[TABLE]
we observe that
[TABLE]
hence from which it follows that .The proof is complete. ∎
Acknowledgments
The authors are grateful to Dr. M. Farrokhi Derakhshandeh Ghouchan for his valuable suggestions and help in carrying out this work.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
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