Groups equal to a product of three conjugate subgroups

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 = (Bn 0 B) 2 = BB n 0 B where n 0 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.

Factorizations of finite groups by conjugate subgroups which are solvable or nilpotent

“…In contrast, if G is a finite nonsolvable group, then γ cp (G) = 3 ([8]). Moreover, in [8] we proved that γ cp (G) = 3 holds for any group G with a BN -pair and a finite Weyl group, and this optimal result arises from choosing A to be the Borel subgroup of G which is in particular solvable. This motivated us to diversify the analysis of cp-factorizations in the present paper by imposing conditions on the subgroup A.…”

“…In this paper we continue the study of minimal length factorizations of (mainly finite) groups into products of conjugate subgroups, that was initiated in [17] and [8]. For a group G and A ≤ G, a conjugate product factorization (a cp-factorization) of length k of G by A, is a factorization G = A 1 • • • A k where A 1 , .…”

Products of Sylow Subgroups in Suzuki and Ree Groups

“…Apart from the decomposition into a product of Sylow subgroups (alias unitriangular factorization), the articles [4,6] also study a decomposition into a product of the conjugates of the Borel subgroup, which is also known as the Gauss decomposition [11]. We now list some remarks concerning their generalizations:…”

Dioid partitions of groups

“…Its first systematic treatment was carried by H. Wielandt [28], and since then it has been further developed and found fruitful applications (see [18], [19]). Our interest in studying Schur dioids was motivated by an observation in [7]. 1 In the first part of the paper we consider general properties of d-partitions.…”

“…Then Π <A is a d-partition of A with e Π<A = e Π and Π ′ := {AπA|π ∈ Π\Π <A } ∪ {A} is a d-partition of G with e Π ′ = A. 1 In [7] it is proved that any group with a BN -pair and a finite Weyl group is the product of three conjugates of the Borel subgroup B [7, Theorem 1.5]. The calculations needed for the proof take place in the Schur dioid induced by the double cosets of B.…”