Embedding Permutation Groups Into Wreath Products in Product Action

Abstract

We consider the wreath product of two permutation groups G ≤ Sym Γ and H ≤ Sym ∆ as a permutation group acting on the set Π of functions from ∆ to Γ. Such groups play an important role in the O'Nan-Scott theory of permutation groups and they also arise as automorphism groups of graph products and codes. Let X be a subgroup of Sym Γ Sym ∆. Our main result is that, in a suitable conjugate of X, the subgroup of SymΓ induced by a stabiliser of a coordinate δ ∈ ∆ only depends on the orbit of δ under the induced action of X on ∆. Hence, if X is transitive on ∆, then X can be embedded into the wreath product of the permutation group induced by the stabiliser X δ on Γ and the permutation group induced by X on ∆. We use this result to describe the case where X is intransitive on ∆ and offer an application to error-correcting codes in Hamming graphs.2010 Mathematics subject classification: primary 05C25; secondary 20B05, 20B25, 20B35, 20D99.

Inclusions of innately transitive groups into wreath products in product action with applications to 2-arc-transitive graphs

“…As both G and M are transitive on Ω, the following is a direct consequence of Theorems 1.1-1.2 of [PS12].…”

“…The aim of this section is to combine the theory of cartesian decompositions with the Embedding Theorem in [PS12] to derive useful facts on the inclusion of G into W that can be used in the characterization of (G, 2)-arc-transitive graphs in Sections 3-5. The main results do not refer to E directly, but the fact that W and G preserve E plays a central role in their proofs.…”

“…Combining this with [BPS04, Proposition 2.1] gives the following. In [PS12], we defined, for j ∈ ℓ, the j-th component of the group G. Let us express this component in terms of the inclusion G W . Let W j denote the stabilizer of j under π.…”

Classification of a family of completely transitive codes

“…n 2 and (n, r) = (2, 2), (2, 3) Table 2. Possible X and m in Almost Simple Case g ∈ Aut(Γ) such that X g X Q 1 wr µ(X) [22]. The group X g is of almost simple type, satisfies Hypothesis 1 of [9], and is faithful on B .…”

Locally s-arc-transitive graphs arising from product action

“…, 58, 47,10,15, 70, 62,13,32, 59, 57,31, 66,22,24, 67, 48,27,35, 50, 45,12,23,11, 52,4, 64, GAP, we see thatL ∼ = C 71 :C 70 × C 9 , R ∼ = C 19 :C 18 × C 35 , L ∩ R ∼ = C 630 , L, R = G,and by Lemma 3.1, the coset graph Cos(G, L, R) is a connected locally (G, 2)-arc-transitive graph.…”