Fischer group Fi22 - Knowlpedia

The Fischer group Fi22 $\mathrm{Fi}_{22}$ is the group with generators $a,b$ and the following complete defining relations:

\begin{gathered} a^2=b^{13}=(ab)^{11}=(ab^2)^{21}=1,\\ [a,b]^3=[a,b^2]^3=[a,b^3]^3=[a,b^4]^2=[a,b^5]^3=1,\\ [a,bab^2]^3=[a,b^{-1}ab^{-2}]^2=1,\\ [a,bab^5]^2=[a,b^2ab^5]^2=1. \end{gathered}

Here $[x,y]=x^{-1}y^{-1}xy$ is the commutator. The presented group consists of words in $a,b$ and their inverses, identified by these relations; multiplication is concatenation followed by this identification.

Order and structure

This is a nonabelian finite simple group, with

|\mathrm{Fi}_{22}|=64561751654400=2^{17}\cdot3^{9}\cdot5^{2}\cdot7\cdot11\cdot13.

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.

Reading the presentation

Every displayed equality is imposed on the free group on the two generators. This is a full presentation; the shorter generator-order tests on the group’s main ATLAS page are only a semi-presentation. The identification with the named sporadic simple group is the cited presentation result.

The displayed version uses $(ab^2)^{21}=1$ . The downloadable Magma file documents that this may instead be replaced by $(ababab^{-3})^5=1$ ; the two variants must not be confused with omitting both relations.

References

ATLAS, Fischer group Fi22, order heading.
ATLAS, full presentation F22G1-P1, displayed relation list and Magma source.