Simple Twisted Group Algebras of Dimensionp⁴and their Semi-Centers

Abstract

For simple twisted group algebra over a group G, if G is Hall subgroup of G then the semi-center is simple. Simple twisted groups algebras correspond to groups of central type. We classify all groups of central type of order p 4 where p is prime and use this to show that for odd primes p there exists a unique group G of order p 4 such that there exists simple twisted group algebra over G with a commutative semi-center. Moreover, if 1 < |G| < 64, then the semi-center of simple twisted group algebras over G is non-commutative and this bounds are strict.

Exotic Coactions

“…The smallest nonabelian examples have order 16, where there are two nonisomorphic groups of central type. In Example 6.7 we will discuss a couple of examples that we found in [Sch16].…”