Some Extensions of PSL(2, p²) are Uniquely Determined by Their Complex Group Algebras

Abstract

In Tong-Viet's, 2012 work, the following question arose: Question. Which groups can be uniquely determined by the structure of their complex group algebras?

It is proved here that some simple groups of Lie type are determined by the structure of their complex group algebras. Let p be an odd prime number and S = PSL(2, p ²). In this paper, we prove that, if M is a finite group such that S < M < Aut(S), M = ℤ₂ × PSL(2, p ²) or M = SL(2, p ²), then M is uniquely determined by its order and some information about its character degrees. Let X ₁(G) be the set of all irreducible complex character degrees of G counting multiplicities. As a consequence of our results, we prove that, if G is a finite group such that X ₁(G) = X ₁(M), then G ≅ M. This implies that M is uniquely determined by the structure of its complex group algebra.

Key Words:

• Character degrees
• Order
• Projective special linear group

2010 Mathematics Subject Classification:

• 20C15
• 20D05
• 20D60

ACKNOWLEDGMENTS

The authors would like to thank the referee for very useful suggestions which improved the results. The authors are very thankful to Professor J. Shareshian for his guidance in the proof of Lemma 2.6. This paper is dedicated to our parents, Professor Amir Khosravi and Soraya Khosravi.

Notes

Communicated by J. Zhang.