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- Title
On noninner automorphisms of finite p-groups that fix the Frattini subgroup elementwise.
- Authors
Ghoraishi, S. Mohsen
- Abstract
Let G be a finite p-group and let L⋆(G)={a∈Z(Φ(G))|a2p∈Z(G)}. In this paper we show that if L⋆(G) lies in the second center Z2(G) of G, then G admits a noninner automorphism of order p, when p is an odd prime, and order 2 or 4, when p=2. Moreover, the automorphism can be chosen so that it induces the identity on the Frattini subgroup Φ(G). When p>2, this reduces the verification of the well-known conjecture that states every finite nonabelian p-group G admits a noninner automorphism of order p to the case in which Z2⋆(G)≨L⋆(G),andCG(Z2⋆(G))=Φ(G), where Z2⋆(G)={a∈Z2(G)|ap∈Z(G)}. In addition, it follows that if G is a finite nonabelian p-group, p≥2, such that Z(Φ(G)) is a cohomologically trivial G/Φ(G)-module, then G satisfies the above mentioned condition, and as a consequence we show that the order of G is at least p8.
- Subjects
FINITE groups; AUTOMORPHISM groups; FRATTINI subgroups; NONABELIAN groups; AUTOMORPHISMS
- Publication
Journal of Algebra & Its Applications, 2018, Vol 17, Issue 7, pN.PAG
- ISSN
0219-4988
- Publication type
Article
- DOI
10.1142/S0219498818501372