L-balance theorem

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
Here L_2′ 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.
A consequence is that if a'' and b are commuting involutions of a group G then
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.