L-balance theorem

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 L2′(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).

References