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On Autocentral Automorphisms of Finite p-Groups
Results in Mathematics ( IF 1.1 ) Pub Date : 2021-01-09 , DOI: 10.1007/s00025-020-01336-8
Sandeep Singh Chahal , Deepak Gumber , Hemant Kalra

Let G be a finite group. An automorphism $$\alpha $$ α of G is called an $$\text{ IA }$$ IA automorphism if it induces the identity automorphism on the abelianized group $$G/G'$$ G / G ′ . Let $$\text{ IA }(G)$$ IA ( G ) be the group of all $$\text{ IA }$$ IA automorphisms of G , and let $$\text{ IA }(G)^*$$ IA ( G ) ∗ be the group of all $$\text{ IA }$$ IA automorphisms of G which fix Z ( G ) element-wise. Let $$\mathrm {Var}(G)$$ Var ( G ) be the group of all autocentral automorphisms of G . We find necessary and sufficient conditions for a finite p -group G such that $$\text{ IA }(G)^*=\text{ Var }(G)$$ IA ( G ) ∗ = Var ( G ) . We also extend the famous result of Adney and Yen (Theorem 1 of Ill J Math 9:137–143, 1965).

中文翻译:

关于有限p群的自中心自同构

令 G 为有限群。G 的自同构 $$\alpha $$ α 称为 $$\text{ IA }$$ IA 自同构,如果它在阿贝尔化群 $$G/G'$$ G / G ′ 上引入恒等自同构。令 $$\text{ IA }(G)$$ IA ( G ) 是 G 的所有 $$\text{ IA }$$ IA 自同构的群,令 $$\text{ IA }(G)^* $$ IA ( G ) ∗ 是 G 的所有 $$\text{ IA }$$ IA 自同构的群,它们按元素固定 Z ( G )。令 $$\mathrm {Var}(G)$$ Var ( G ) 是 G 的所有自中心自同构的群。我们找到有限 p 群 G 的充分必要条件,使得 $$\text{ IA }(G)^*=\text{ Var }(G)$$ IA ( G ) ∗ = Var ( G ) 。我们还扩展了 Adney 和 Yen 的著名结果(Ill J Math 9:137–143, 1965 的定理 1)。
更新日期:2021-01-09
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