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Unique conjugator factorization restricts to a normalizer splitting

Statement

Let $A$ be a group with specified subgroups $A_{I}$ . Form the groupoid $C$ with arrows $(I, g, J)$ satisfying $g^{- 1} A_{I} g = A_{J}$ , and product $(I, g, J) (J, h, K) = (I, g h, K)$ . Let $P$ have only vertex arrows $A_{I}$ , and let $R \subseteq C$ be a wide subgroupoid. Suppose every arrow $g : I \to J$ has a unique factorization $g = p r$ with $p \in A_{I}$ and $r \in R (I, J)$ . Then $C = P ⋊ R$ in the internal sense, and $N_{A} (A_{I}) = A_{I} ⋊ R_{I}$ . Explicitly, every normalizer element has a unique such factorization, $A_{I} \cap R_{I} = {1}$ and the action in the usual product coordinates is $α_{r} (q) = r q r^{- 1}$ .

Facts & Assumptions

[F1]

The set-sized groupoid axioms, tagged arrows, vertex groups and left-to-right product are specified in Set-sized groupoids, functors and composition conversion.

[F2]

A wide subgroupoid $R \subseteq C$ has all the objects of $C$ , contains their identity arrows, and is closed under composable products and inverses, with these operations inherited from $C$ . This is an inline definition using the groupoid structure in [F1]. The unique factorization hypothesis is an assumption of this lemma, to be supplied separately in applications. The internal groupoid semidirect notation means unique factorization and $A_{I} r = r A_{J}$ for every $r : I \to J$ .

[F3]

Groups and subgroups use Group and abelian group and Subgroup. The normalizer is $N_{A} (H) = {g \in A : g H g^{- 1} = H}$ as in The normalizer N_G(H)={g∈ G:gHg⁻¹=H} of a subgroup.

[F4]

Normality means invariance under every conjugation by the ambient group, as in Normal subgroup: invariance under conjugation. The internal group semidirect product $G = N ⋊ H$ means $N ⊴ G$ , $G = N H$ and $N \cap H = {1}$ , as in An internal semidirect product and a complement to a normal subgroup.

[F5]

An action by automorphisms satisfies $α_{r s} = α_{r} \circ α_{s}$ and $α_{1} = id$ , as in An action of a group H on a group N by automorphisms. For such an action, the external semidirect product has coordinates $(p, r)$ with multiplication $(p, r) (q, s) = (p α_{r} (q), r s)$ , as in The external semidirect product N⋊_α H.

Proof

Given: The group, subgroup family, wide subgroupoid and assumed unique factorizations.

Goal: Prove the groupoid and normalizer semidirect conclusions.

1.1F1F3algebra

If $g : I \to J$ and $h : J \to K$ , then $(g h)^{- 1} A_{I} (g h) = h^{- 1} (g^{- 1} A_{I} g) h = A_{K}$ . Identity arrows and inverse labels $g^{- 1}$ have the required endpoints. Associativity follows from the group multiplication. Thus the tagged arrows form the groupoid in [F1]. At $I$ , the arrow condition $g^{- 1} A_{I} g = A_{I}$ is equivalent to $g A_{I} g^{- 1} = A_{I}$ , so its vertex group is $N_{A} (A_{I})$ under the identification $(I, g, I) \mapsto g$ . Also $P$ is a subgroupoid because each $A_{I}$ is a subgroup.

2.1F2F3F4step 1.1

An arrow $r : I \to J$ satisfies $r^{- 1} A_{I} r = A_{J}$ , equivalently $A_{I} r = r A_{J}$ . Along with the assumed unique factorization this is exactly the specified internal groupoid semidirect property. By the closure and identity requirements in [F2], $R_{I} = R (I, I)$ is a subgroup of $N_{A} (A_{I})$ . At a vertex, factorization gives $N_{A} (A_{I}) = A_{I} R_{I}$ . An element common to $A_{I}$ and $R_{I}$ has the two factorizations $x \cdot 1$ and $1 \cdot x$ , so uniqueness implies $x = 1$ . Finally $A_{I}$ is normal in $N_{A} (A_{I})$ by the normalizer definition. Hence these are the internal group semidirect conditions of [F4].

3.1F3F5step 2.1algebra∎

For $r \in R_{I}$ , conjugation $α_{r} : q \mapsto r q r^{- 1}$ preserves $A_{I}$ , preserves multiplication, and has inverse conjugation by $r^{- 1}$ , so is an automorphism. Also $α_{r r^{'}} = α_{r} \circ α_{r^{'}}$ and $α_{1} = {id}_{A_{I}}$ , giving the action in [F5]. The map $(p, r) \mapsto p r$ is bijective by factorization and satisfies $(p r) (q s) = p (r q r^{- 1}) r s$ , exactly the external semidirect group law of [F5]. This proves all vertex claims.

Depends on

Used by · 1

Independent spherical ribbon splitting input

Dependency tree · 2 levels

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