A module-theoretic approach to abelian automorphism groups

Abstract

Abstract. There are several examples in the literature of finite non-abelian p-groups whose automorphism group is abelian. For some time only examples that were special p-groups were known, until Jain and Yadav [JY12] and Jain, Rai and Yadav [JRY13] constructed several non-special examples. In this paper we show how a simple module-theoretic approach allows the construction of non-special examples, starting from special ones constructed by several authors, while at the same time avoiding further direct calculations.

On the Conjugacy Search Problem in Extraspecial $p$-Groups

“…Further, central products with the efficient Cdecomposability property that we introduce must be avoided, and a chosen platform must in a way be "atomic". This is practically relevant even beyond just p-groups, since several nonabelian groups are constructed by combining smaller groups by taking direct, central, and semidirect products (see, for example, [2], [4]). Finally, our results also show a polynomial time solution for the CDP in any extraspecial p-group.…”

“…The results on central products also demonstrate that while considering any platform group for a CSP-based system, care must be taken to ensure that an efficient decomposition into a central product is not possible. This is practically significant for future work in group-based cryptography since several nonabelian groups, and in particular, p-groups, are constructed by combining smaller p-groups by taking direct, central, and semidirect products (see, for example, [2], [4]).…”

Finite $p$-groups of class two with a small multiple holomorph

“…Here V is a finite dimensional vector space over F p and Λ 2 V denotes its exterior square. The construction is from [Car15], to be reviewed in Section 3. Let us note that the constructed p-groups G have order…”

“…Let p be an odd prime and let n ≥ 2 be an integer. Let us review a construction, via linear algebra as described in [Car15], of p-groups of order p n+( n 2 ) and class two. Let V be an n-dimensional vector space over F p and let Λ 2 V denote its exterior square.…”