Representation theoretic existence proof for Fischer group Fi_{23}

TL;DR

This paper provides a new representation theoretic and algorithmic proof for the existence of Fischer's sporadic simple group Fi_{23}, including explicit constructions and computational results confirming its properties.

Contribution

The paper introduces new algorithms for finite permutation groups and offers a novel existence proof for Fi_{23} using representation theory and computational methods.

Findings

Constructed the three non-isomorphic extensions E_i of M_{23} over GF(2)

Successfully embedded G inside GL_{782}(17) and computed its character table

Confirmed that G and Fi_{23} share the same character table

Abstract

In the first section of this senior thesis the author provides some new efficient algorithms for calculating with finite permutation groups. They cannot be found in the computer algebra system MAGMA, but they can be implemented there. For any finite group G with a given set of generators, the algorithms calculate generators of a fixed subgroup of G as short words in terms of original generators. Another new algorithm provides such a short word for a given element of G. In the later sections, the author gives a self-contained existence proof for Fischer's sporadic simple group Fi_{23} using G. Michler's Algorithm [11] constructing finite simple groups from irreducible subgroups of GL_n(2). This sporadic group was originally discovered by B. Fischer in [6] by investigating 3-transposition groups, see also [5]. This thesis gives a representation theoretic and algorithmic existence proof for his group. The author constructs the three non-isomorphic extenstions E_i by the two 11-dimensional non-isomorphic simple modules of the Mathieu group M_{23} over F=GF(2). In two cases Michler's Algorithm fails. In the third case the author constructs the centralizer H=C_G(z) of a 2-central involution z of E_i in any target simple group G. This allows the author to construct G inside GL_{782}(17). Its four generating matrices, character table and representatives for conjugacy classes are computed. It follows that G and Fi_{23} have the same character table.