On a conjecture about automorphisms of finite \(p\)-groups. (Q1042424)
From MaRDI portal
 | This is the item page for this Wikibase entity, intended for internal use and editing purposes. Please use this page instead for the normal view: On a conjecture about automorphisms of finite \(p\)-groups. |
scientific article; zbMATH DE number 5646236
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | On a conjecture about automorphisms of finite \(p\)-groups. |
scientific article; zbMATH DE number 5646236 |
Statements
On a conjecture about automorphisms of finite \(p\)-groups. (English)
0 references
14 December 2009
0 references
The main result of this note asserts that if \(G\) is a finite nonabelian \(p\)-group satisfying any of the following conditions: 1) \(\text{rank}(G'\cap Z(G))\neq\text{rank}(Z(G))\), 2) \(Z(\text{Inn}(G))\) is cyclic, 3) \((Z_2(G)\cap Z(\Phi(G)))/Z(G)\) is not elementary Abelian of rank \(rs\), where \(r=d(G)\), \(s=\text{rank}(Z(G))\), then \(G\) has a noninner automorphism of order \(p\) fixing \(\Phi(G)\) elementwise.
0 references
finite \(p\)-groups
0 references
noninner automorphisms
0 references
central automorphisms
0 references
