On the Twisted Group Ring Isomorphism Problem for isoclinic groups

Abstract

In this article, we consider a general version of the classical group ring isomorphism problem, called the twisted group ring isomorphism problem (TGRIP), which determines an isomorphism between the twisted complex group algebras of finite groups. Although for isoclinic groups $G$ and $H$ their complex group algebras are isomorphic, i.e, $\mathbb{C}G\cong \mathbb{C}H$, their twisted complex group algebras need not be isomorphic. Continuing this line of investigation, we establish sufficient conditions under which two isoclinic groups have isomorphic twisted complex group algebras, thereby providing solutions to (TGRIP) for isoclinic groups. As applications, we study (TGRIP) for special $p$-groups of rank $2$, unicentral groups, and nilpotent groups of class $2$ with elementary abelian Schur multipliers, all considered up to isoclinism. We also present several examples illustrating the main results, including a complete solution of (TGRIP) for special $p$-groups of rank $2$ of order $p^6(p\ge 3)$