The modular group algebra problem for groups of orderp⁵

Abstract

We show that p-groups of order p 5 are determined by their group algebras over the field of p elements. Many cases have been dealt with in earlier work of ourselves and others. The only case whose details remain to be given here is that of groups of nilpotency class 3 for p odd.

Isomorphism problem of Unitary Subgroups of Group Algebras

“…The most general result of the isomorphism problem related to abelian basic groups can be found in [21]. For non-abelian group algebras, this problem was investigated in [1,2,3,18,22,23,24,26,27] and [28]. For an overview we recommend the survey paper [8].…”

Computing automorphism groups and testing isomorphisms for modular group algebras

“…This problem is still open, despite various efforts towards proving the claim or finding counterexamples to it. The claim has been proved, for example, for abelian p-groups [4], p-groups of class 2 and exponent p [17], metacyclic p-groups [22] and groups of order p n dividing 2 7 [2,28] or p 5 [21].…”

A cohomological approach to the modular isomorphism problem

“…• abelian p-groups (Deskins 1956, see [6]), • p-groups of order 6 p 4 (Passman 1965, see [14]), • groups of class 2 and exponent p (Passi and Sehgal 1972, see [13]), • groups of order 2 5 (Makasikis 1976, faults rectiÿed by Sandling 1984, see [17]), • metacyclic p-groups (completed by Sandling 1994, see [18]), • groups of order p 5 (completed by Salim and Sandling 1994, see [16]), • groups of order 2 6 (Wursthorn 1994, see [20]), • groups of order 2 7 (Bleher, Kimmerle, Roggenkamp and Wursthorn 1997, see [3]). …”