# On finite $p$-groups with abelian automorphism group

Source: [On finite $p$-groups with abelian automorphism group](<https://www.sota2.com/research/paper/68fa54fb1e209131ce99007f>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-25T13:18:36.524Z

Authors: Vivek K. Jain, Pradeep K. Rai, Manoj K. Yadav

Published: 2013-04-07T09:55:42.000Z

## 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 $\\gamma\_2(G) \< \\mathrm{Z}(G) \< \\Phi(G)$, where $\\gamma\_2(G)$, $\\mathrm{Z}(G)$ 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) $\\mathrm{Z}(G) = \\Phi(G)$ 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.

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/1304.1974>)

[PDF](<https://arxiv.org/pdf/1304.1974.pdf>)