Representations of extensions of simple groups.

Area of Science:

• Group Theory
• Representation Theory
• Abstract Algebra

Background:

• Feit and Tits (1978) established a key result concerning minimal dimensional projective representations of finite extensions of nonabelian simple groups.
• Kleidman and Liebeck (1989) identified specific exceptions related to groups of Lie type in characteristic 2.

Purpose of the Study:

• To generalize the Feit-Tits theorem to all low-dimensional projective representations, not solely minimal ones.
• To extend the theorem's applicability from finite nonabelian simple groups to all characteristically simple groups.

Main Methods:

• Analysis of projective representations in group theory.
• Generalization of existing theorems in abstract algebra.
• Examination of group extensions and characteristically simple groups.

Main Results:

• The study confirms that low-dimensional projective representations of finite extensions of characteristically simple groups generally factor through the base group.
• Identified specific conditions and exceptions where this factorization does not hold, expanding on prior work.

Conclusions:

• The findings provide a more comprehensive understanding of projective representations in a broader class of groups.
• This research contributes to the ongoing development of representation theory for finite groups.