L-balance theorem
In mathematical finite group theory, the L-balance theorem was proved by (Gorenstein Walter). 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
- [math]\displaystyle{ L_{2'}(C_X(T)) \le L_{2'}(X) }[/math]
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
- [math]\displaystyle{ L_{2'}(L_{2'}(C_a)\cap C_b) = L_{2'}(L_{2'}(C_b)\cap C_a) }[/math]
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 Lp-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
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