L-balance theorem

In mathematical finite group theory, the L-balance theorem was proved by Gorenstein & Walter (1975). 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

${\displaystyle L_{2'}(C_{X}(T))\leq L_{2'}(X)}$

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

${\displaystyle L_{2'}(L_{2'}(C_{a})\cap C_{b})=L_{2'}(L_{2'}(C_{b})\cap C_{a})}$

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

• Gorenstein, D.; Walter, John H. (1975), "Balance and generation in finite groups", Journal of Algebra, 33: 224–287, doi:10.1016/0021-8693(75)90123-4, ISSN 0021-8693, MR 0357583