L-balance theorem

Template:Short description In mathematical finite group theory, the L-balance theorem was proved by Template:Harvtxt. 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

$L_{2^{\prime }}(C_X(T))\le L_{2^{\prime }}(X)$

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

$L_{2^{\prime }}(L_{2^{\prime }}(C_a)\cap C_b)=L_{2^{\prime }}(L_{2^{\prime }}(C_b)\cap C_a)$

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

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