On unit group of finite semisimple group algebras of non-metabelian groups up to order 72

Abstract

. We characterize the unit group of semisimple group algebras F q G of some non-metabelian groups, where F q is a field with q = p k elements for p prime and a positive integer k . In particular, we consider all 6 non-metabelian groups of order 48, the only non-metabelian group (( C 3 × C 3 ) ⋊ C 3 ) ⋊ C 2 of order 54, and 7 non-metabelian groups of order 72. This completes the study of unit groups of semisimple group algebras for groups upto order 72.