L-balance theorem explained

In mathematical finite group theory, the L-balance theorem was proved by .The letter L stands for the layer of a group, and "balance" refers to the property discussed below.

Statement

The L-balance theorem of Gorenstein and Walter states that if X is a finite group and T a 2-subgroup of X then

L _{2'} (C _{X(T))} \le L _{2'} (X)

Here L_2′(X) stands for the 2-layer of a group X, which is the product of all the 2-components of the group, the minimal subnormal subgroups of X mapping onto components of X/O(X).

A consequence is that if a and b are commuting involutions of a group G then

L _{2'} (L _{2'} (C _{a)\cap} C _{b)} = L _{2'} (L _{2'} (C _{b)\cap} C _{a)} This is the property called L-balance.

More generally similar results are true if the prime 2 is replaced by a prime p, and in this case the condition is called L_p-balance, but the proof of this requires the classification of finite simple groups (more precisely the Schreier conjecture).