Groups equal to a product of three conjugate subgroups

2015/01/22 by John Cannon, Martino Garonzi, Cannon, John +7
Mathematics · #FOS: Mathematics #Group Theory (math.GR) #math.GR

paper · pdf · doi:10.48550/arxiv.1501.05676

arxiv created 2015/01/22 · arxiv updated 2015/01/26

Abstract

Let G be a finite non-solvable group. We prove that there exists a proper subgroup A of G such that G is the product of three conjugates of A, thus replacing an earlier upper bound of 36 with the smallest possible value. The proof relies on an equivalent formulation in terms of double cosets, and uses the following theorem which is of independent interest and wider scope: Any group G with a BN-pair and a finite Weyl group W satisfies G=( Bn₀B) ²=BB^n₀B where n₀ is any preimage of the longest element of W. The proof of the last theorem is formulated in the dioid consisting of all unions of double cosets of B in G. Other results on minimal length product covers of a group by conjugates of a proper subgroup are given.

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