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Theory of Groups (MATH 6270) Decompositions of Permutation Groups We will discuss how a permutation group acting on a finite set can be decomposed into primitive components. There are two kinds of decompositions involved: (i) decomposing an intransitive permutation group into its ‘transitive constituents’, and (ii) decomposing a transitive, but imprimitive permutation group into ‘smaller’ permutation groups, one acting on a nontrivial group, the other on the set of translates of that block. Q If {Ωi : 1 ≤ i ≤ n} isSa partition of a set Ω, and Gi ≤ Sym(Ωi ) for each i, then 1≤i≤n Gi acts faithfully on Ω = 1≤i≤n Ωi by def α · (gi )1≤i≤n = α · gj This action allows us to consider for the unique j (1 ≤ j ≤ n) with α ∈ Ωj . Q 1≤i≤n Gi as a subgroup of Sym(Ω). Lemma 1. Let {Ωi : 1 ≤ i ≤ n} be a partition of a set Ω. Then Q the permutation group Q 1≤i≤n Sym(Ωi ) (acting on Ω) has orbits Ωi (1 ≤ i ≤ n); moreover, 1≤i≤n Sym(Ωi ) is the largest subgroup of Sym(Ω) admitting these sets as orbits. For a permutation group G ≤ Sym(Ω) and an orbit O of G the restriction of G to O is the group G|O = {g|O : g ∈ G} ≤ Sym(O). Theorem 2. If G ≤ Sym(Ω) permutation group with orbits Oi (1 ≤ i ≤ n), Q is an intransitive Q then G is a subgroup of 1≤i≤n G|Oi ≤ 1≤i≤n Sym(Oi ) ≤ Sym(Ω) . Now let G ≤ Sym(Ω) and H ≤ Sym(Λ) where Ω and Λ are finite. Recall1 that the wreath product H o G is defined as the semidirect product H Ω o G determined by the following action of G on the base group H Ω : for x ∈ G and f = (fα )α∈Ω ∈ H Ω , f x ∈ H Ω is defined by f x = (fα·x−1 )α∈Ω for all α ∈ Ω. Furthermore, H o G acts faithfully on Ω × Λ by (α, λ) · (x, f ) = (α · x, λ · fα·x ) for all α ∈ Ω, λ ∈ Λ, x ∈ G, f ∈ H Ω . This action, which is usually referred to as the standard action of H o G, allows us to consider H o G as a subgroup of Sym(Ω × Λ). Lemma 3. Let Ω and Λ be finite sets. (1) The following conditions on a permutation g ∈ Sym(Ω × Λ) are equivalent: (i) g ∈ Sym(Λ) o Sym(Ω); (ii) for each α ∈ Ω there exists β ∈ Ω such that g maps {α} × Λ onto {β} × Λ. (2) In other words, the sets {α} × Λ (α ∈ Ω) are blocks of the permutation group Sym(Λ) o Sym(Ω); moreover, Sym(Λ) o Sym(Ω) is the largest subgroup of Sym(Ω × Λ) admitting these sets as blocks. (3) Consequently, Sym(Λ) o Sym(Ω) acts imprimitively on Ω × Λ, if |Ω|, |Λ| ≥ 2. 1See the handout “Actions of Semidirect Products by Permutations”. 1 2 The next theorem shows that the wreath product of two permutation groups (with its standard action) captures the ‘typical’ imprimitive action. For a permutation group G ≤ Sym(Ω) and a subset ∆ ⊆ Ω, Gk∆ will denote the following subgroup of Sym(∆): Gk∆ = {g|∆ ∈ G : g ∈ G, ∆ · g = ∆}. Theorem 4. Let G ≤ Sym(Ω) be a transitive but imprimitive permutation group on a finite ˆ = {∆·g : g ∈ G}, then there exists an embedding set Ω. If ∆ is a nontrivial block of G and ∆ ˆ ≤ Sym(∆ ˆ × ∆) ϕ : G → Gk∆ o Ĝ ≤ Sym(∆) o Sym(∆) ˆ Moreover, the action of G on Ω is permutation where Ĝ is the natural action of G on ∆. ˆ × ∆ (if we identify G with isomorphic to the action of the subgroup ϕ(G) of Gk∆ o Ĝ on ∆ ϕ(G) via ϕ). ˆ such that Sketch of proof. Choose a transversal {gi : 1 ≤ i ≤ n} for the blocks in ∆ ˆ = {∆ · gi : 1 ≤ i ≤ n} with ∆ = ∆ · g1 . The mapping g1 = 1; then ∆ ˆ × ∆ → Ω, ι: ∆ (∆ · gi , α) 7→ α · gi ˆ Hence is a bijection, which maps the sets {∆ · gi } × ∆ (1 ≤ i ≤ n) onto the blocks ∆ · gi in ∆. ˆ × ∆), g 7→ ιgι−1 maps G the (permutation isomorphic) group embedding ψ : G → Sym(∆ ˆ × ∆) for which the sets {∆ · gi } × ∆ (1 ≤ i ≤ n) are blocks. onto a subgroup ψ(G) of Sym(∆ ˆ It is straightforward to Therefore, by Lemma 3, ψ(G) is a subgroup of Sym(∆) o Sym(∆). check that, in fact, ψ(G) is a subgroup of Gk∆ o Ĝ. With the same notation as in Theorem 4, let α ∈ ∆. By the 1–1 correspondence between the blocks of G containing α and the groups K satisfying Gα ≤ K ≤ G, we see ˆ is permutation isomorphic to the (faithful) acthat the permutation group Ĝ ≤ Sym(∆) tion of G/coreG (K) on {Kx : x ∈ G}, while the permutation group Gk∆ ≤ Sym(∆) is permutation isomorphic to the (faithful) action of K on {Gα y : y ∈ K}. Thus, by choosing a maximal block ∆ in Theorem 4, we can embed G (permutation isomorphically) into a wreath product K1 oH1 ≤ Sym(Γ1 ×∆1 ) where H1 acts primitively on Γ1 and K1 acts transitively on ∆1 . Clearly, |Ω| = |Γ1 ||∆1 |, and since ∆1 is a nontrivial block, |∆1 | < |Ω|. If K1 acts imprimitively on ∆1 , we can repeat the decomposition procedure, and continue as long as possible. The procedure will terminate, since Ω is finite. Hence we get Corollary 5. If G ≤ Sym(Ω) is a transitive but imprimitive permutation group on a finite set Ω, then G can be embedded (permutation isomorphically) into an iterated wreath product ((. . . (Hr o Hr−1 ) o . . .) o H2 ) o H1 ≤ Sym ((. . . (Γr × Γr−1 ) × . . .) × Γ2 ) × Γ1 of primitive permutation groups Hi ≤ Sym(Γi ) (1 ≤ i ≤ r). By the discussion preceding Corollary 5, there is a 1–1 correspondence between these decompositions of G and the sequences of subgroups Gα = Kr < · · · < K1 < K0 = G where each Ki is a maximal subgroup of Ki−1 (1 ≤ i ≤ r). It follows that the decomposition in Corollary 5 for a transitive but imprimitive permutation group G ≤ Sym(Ω) into primitive components Hi is not unique. In general, neither the number r nor the family of primitive permutation groups Hi (which are permutation isomorphic to the action of Ki−1 /coreKi−1 (Ki ) on the right cosets of Ki by right multiplication) is uniquely determined by G.