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I heard that Schur multiplier's played important role in classification of finite simple groups. By means of simple example, can one illustrate how the Schur multiplies played their role in the classification? Was it used as just an (isomorphism) invariant for a group?

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Here is one situation in which they arise - there are probably others.

The Fitting subgroup $F(G)$ of a finite solvable group is defined to be its largest normal nilpotent subgroup, and it has the property that $C_G(F(G)) \le F(G)$.

This idea was extended to all finite groups, and the generalized Fitting subgroup $F^*(G)$ is defined as $F(G)E(G)$, where $E(G)$ is the subgroup generated by all subnormal quasisimple subgroups of $G$. It is not easy to motivate this definition, which took a while to evolve, but it turned out to be just what was needed in the classification, and it play a very central role there. This subgroup has the corresponding property $C_G(F^*(G)) \le F^*(G)$ that $F(G)$ has for solvable groups.

A quasisimple group $H$ is a perfect group ($[H,H]=H$) such that $H/Z(H)$ is a nonabelian simple group. So, in a quasisimple group, $Z(H)$ is a quotient of the Schur Multiplier of $H$.

The proof of CFSG is inductive so most of the time, the composition factors of the groups being considered are "known" finite simple groups. So to know the possible structures of the quasisimple groups involved and hence of $E(G)$, it is necessary to know the Schur multipliers of all of the finite simple groups.

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