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.
Statementedit
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 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
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).
Referencesedit
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
April 11, 2024
balance, theorem, mathematical, finite, group, theory, proved, gorenstein, walter, 1975, letter, stands, layer, group, balance, refers, property, discussed, below, statement, editthe, gorenstein, walter, states, that, finite, group, subgroup, then, displaystyl. In mathematical finite group theory the L balance theorem was proved by Gorenstein amp Walter 1975 The letter L stands for the layer of a group and balance refers to the property discussed below Statement editThe L balance theorem of Gorenstein and Walter states that if X is a finite group and T a 2 subgroup of X then L2 CX T L2 X displaystyle L 2 C X T leq L 2 X nbsp 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 L2 L2 Ca Cb L2 L2 Cb Ca displaystyle L 2 L 2 C a cap C b L 2 L 2 C b cap C a nbsp 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 editGorenstein 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 Retrieved from https en wikipedia org w index php title L balance theorem amp oldid 1032018693, wikipedia, wiki, book, books, library,
