The modular group algebra of a central-elementary-by-abelian p-group (Q1095228)

The title refers to a p-group G for which \(\gamma _ 2(G)^ p\gamma _ 3(G)=1\) where \(\gamma _ n(G)\) is the n-th term of the lower central series of G. Two results are proved about the modular group algebra FG, where F is the field of p elements and G is a p-group: FG determines the isomorphism type of \(G/\gamma _ 2(G)^ p\gamma _ 3(G)\); there is a normal complement to G in the group of units of FG if G is central- elementary-by-abelian. Some partial results are stated for the broader class of elementary-abelian-by-abelian p-groups.