Baby Monster group - Knowlpedia

The Baby Monster group $\mathbb B$ is the subgroup generated by the fixed pair of $4370\times4370$ matrices $A,B$ in ATLAS dataset BG1-f2r4370B0:

\mathbb B=\langle A,B\rangle\leq\mathrm{GL}_{4370}(\mathbb F_{2}).

The files BG1-f2r4370B0.m1 and BG1-f2r4370B0.m2 specify $A$ and $B$ , respectively; their entries are part of this concrete definition. Here $\mathbb F_{2}=\mathbb Z/2\mathbb Z$ is the field with 2 elements, $\mathrm{GL}_{4370}(\mathbb F_{2})$ consists of invertible matrices, and the group operation is matrix multiplication over that field. The exact generator files are linked in References.

Order and structure

This is a nonabelian finite simple group, with

\begin{aligned}|\mathbb B|&=2^{41}\cdot3^{13}\cdot5^{6}\cdot7^{2}\cdot11\cdot13\cdot17\\ &\quad{}\cdot19\cdot23\cdot31\cdot47.\end{aligned}

It belongs to the Monster display block of the sporadic groups. This grouping records the ATLAS organization, without asserting a direct embedding into another displayed group.

The concrete matrix model

This gives a faithful action on $\mathbb F_{2}^{4370}$ . The matrix degree $4370$ is distinct from the number of group elements. The two specified matrices determine the group; tests on the orders of unspecified generators would not constitute this definition.

The sporadic subgroups of the Baby Monster include copies of $\mathrm{Fi}_{23}$ and $\mathrm{Th}$ .

References

ATLAS, Baby Monster group, order heading and “Representations.”
ATLAS, matrix model BG1-f2r4370B0, “About this representation” and “Download”; exact matrices A and B.