Simple twisted group algebras of dimension $p^4$ and their semi-centers

Abstract

For simple twisted group algebra over a group $G$, if $G^{\mid }$ 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.