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On finite $p$-groups with abelian automorphism group | Vivek K. Jain
; Pradeep K. Rai
; Manoj K. Yadav
; | Date: |
7 Apr 2013 | Abstract: | We construct, for the first time, various types of specific non-special
finite $p$-groups having abelian automorphism group. More specifically, we
construct groups $G$ with abelian automorphism group such that $gamma_2(G) <
(G) < Phi(G)$, where $gamma_2(G)$, $(G)$ and $Phi(G)$ denote the
commutator subgroup, the center and the Frattini subgroup of $G$ respectively.
For a finite $p$-group $G$ with elementary abelian automorphism group, we show
that at least one of the following two conditions holds true: (i) $(G) =
Phi(G)$ is elementary abelian; (ii) $gamma_2(G) = Phi(G)$ is elementary
abelian, where $p$ is an odd prime. We construct examples to show the existence
of groups $G$ with elementary abelian automorphism group for which exactly one
of the above two conditions holds true. | Source: | arXiv, 1304.1974 | Services: | Forum | Review | PDF | Favorites |
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