Verification of the ordinary character table of the Baby Monster

doi:10.1016/j.jalgebra.2019.06.047

Journal of Algebra

Volume 561, 1 November 2020, Pages 111-130

https://doi.org/10.1016/j.jalgebra.2019.06.047 Get rights and content

Abstract

We prove the correctness of the character table of the sporadic simple Baby Monster group that is shown in the $ATLAS$ of Finite Groups.

Introduction

Jean-Pierre Serre has raised the question of verification of the ordinary character tables that are shown in the $ATLAS$ of Finite Groups [6]. This question was partially answered in the paper [5], the remaining open cases being the largest two sporadic simple groups, the Baby Monster group $B$ and the Monster Group $M$ , and the double cover $2 . B$ of $B$ .

The current paper describes a verification of the character table of $B$ . The computations shown in [3] then imply that also the $ATLAS$ character table of $2 . B$ is correct. As in [5], one of our aims is to provide the necessary data in a way that makes it easy to reproduce our computations.

The $ATLAS$ character table of the Baby Monster derives from the original calculation of the conjugacy classes and rational character table by David Hunt, described very briefly in [9]. The irrationalities were calculated by the CAS team in Aachen [11].

Section snippets

Strategy

We begin with a preliminary section, Section 3, whose aim is to prove that certain specified matrices do indeed generate copies of the Baby Monster. These matrices can then be used in the main computation. The $Y_{555}$ presentation of the BiMonster implies a $Y_{433}$ presentation for $B$ (see [10]), given that the Schur multiplier $H^{2} (M, C^{⁎})$ of the Monster has odd order. The Schur multiplier of the Monster was calculated by Griess [8]. We use the $Y_{433}$ presentation to prove that three pairs of matrices, of

Verifying a presentation for the Baby Monster

In this section we give words in the ‘standard generators’ for the Baby Monster, that represent the 11 transpositions in the $Y_{433}$ presentation. This provides a relatively straightforward test to prove that a given black-box group is in fact isomorphic to the Baby Monster.

Centralizers of prime order elements in the Baby Monster

In this section we determine the classes of prime order elements, and the orders of their centralizers, in the Baby Monster. Much of this information comes from Stroth's 1976 paper [16]. In cases where [16] does not give full information, our strategy is first to use a certified copy of the Baby Monster from [21] to give lower bounds on both the number of conjugacy classes and the orders of the respective centralizers, and then to use local arguments, together with information about the

Obtaining the class list

Our strategy for obtaining the list of conjugacy classes in the Baby Monster is first to determine the classes of even order elements, by computing the character tables of subgroups containing the four distinct involution centralizers, and noting down the conjugacy classes of elements in each subgroup that power to the relevant involution class. (The centralizers of involutions in classes 2A, 2B, 2C are in fact maximal, although it is not necessary to know this, so we have no choice but to use

Computing the irreducible characters of the Baby Monster

From the previous sections, we know that $B$ contains subgroups of the structures $2 .^{2} E_{6} (2) . 2$ , ${Fi}_{23}$ , and $HN . 2$ . The ordinary character tables of these groups have been verified (see [5]) and thus may be used in our computations. The class fusions from these subgroups to $B$ can be computed with the methods available in GAP [7]. Moreover, in Section 6.1, we have computed the character table of the 2B-centralizer in $B$ . The class fusion from $2^{1 + 22} . {Co}_{2}$ to $B$ is determined by evaluating the three

Acknowledgements

We thank Chris Parker for significant contributions to the original version of this paper, and we thank the referee for helpful comments that enabled us to avoid the need for them.

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Verification of the ordinary character table of the Baby Monster

Abstract

Introduction

Section snippets

Strategy

Verifying a presentation for the Baby Monster

Centralizers of prime order elements in the Baby Monster

Obtaining the class list

Computing the irreducible characters of the Baby Monster

Acknowledgements

J. Symbolic Comput.

J. Algebra

J. Algebra

J. Algebra

J. Algebra

J. Algebra

Constructing the ordinary character tables of some Atlas groups using character theoretic methods

Some steps in the verification of the ordinary character table of the Baby Monster group