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

TL;DR

This paper classifies groups of order p^4 of central type and analyzes the structure of simple twisted group algebras over these groups, revealing conditions for the semi-center's simplicity and commutativity.

Contribution

It provides a complete classification of central type groups of order p^4 and characterizes when their simple twisted group algebras have commutative semi-centers.

Findings

Unique group of order p^4 with simple twisted algebra and commutative semi-center for odd p.

Semi-center is non-commutative for groups of order less than 64.

Classification of all groups of central type of order p^4.

Abstract

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