Thompson sporadic group

The Thompson sporadic group $\mathrm{Th}$ is the subgroup generated by the fixed pair of $248\times248$ matrices $A,B$ in ATLAS dataset ThG1-f2r248B0:

\mathrm{Th}=\langle A,B\rangle\leq\mathrm{GL}_{248}(\mathbb F_{2}).

The files ThG1-f2r248B0.m1 and ThG1-f2r248B0.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, $\mathrm{GL}_{248}(\mathbb F_{2})$ 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

|\mathrm{Th}|=90745943887872000=2^{15}\cdot3^{10}\cdot5^{3}\cdot7^{2}\cdot13\cdot19\cdot31.

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 $\mathbb F_{2}^{248}$ . The matrix degree $248$ 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 notation $\mathrm{Th}$ denotes the finite sporadic group; Thompson’s infinite groups $F,T,V$ are different objects.

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

References

ATLAS, Thompson sporadic group, order heading and “Representations.”
ATLAS, matrix model ThG1-f2r248B0, “About this representation” and “Download”; exact matrices A and B.