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Transfer is independent of the transversal

Statement

Let $G$ be a finite group, let $H \leq G$ be a subgroup, let $A$ be an abelian group written multiplicatively, and let $φ : H \to A$ be a homomorphism (Transfer homomorphism for a finite index subgroup). Let ${t_{α}}_{α \in H \ G}$ and ${t_{α}^{'}}_{α \in H \ G}$ be two choices of representatives of the right cosets of $H$ in $G$ , and let

$V (x) : = \prod_{α \in H \ G} φ (t_{α} x t_{α x}^{- 1}), V^{'} (x) : = \prod_{α \in H \ G} φ (t_{α}^{'} x t_{α x}^{'}^{- 1})$

be the two products formed from them, where $α x : = H t_{α} x$ . Then $V (x) = V^{'} (x)$ for every $x \in G$ . In particular the transfer $V_{φ}$ is a well-defined function $G \to A$ depending only on $φ$ .

Facts & Assumptions

Given: A finite group $G$ , a subgroup $H \leq G$ , an abelian group $A$ , a homomorphism $φ : H \to A$ , and two transversals ${t_{α}}$ , ${t_{α}^{'}}$ of the right cosets $α \in H \ G$ as in Transfer homomorphism for a finite index subgroup.

[F1]

For every $α$ the coset $α x = H t_{α} x$ is a right coset of $H$ , the assignment $α \mapsto α x$ is a permutation of $H \ G$ , each $t_{α} x t_{α x}^{- 1}$ lies in $H$ , and the finite product $\prod_{α \in H \ G} a_{α}$ of elements of the abelian group $A$ is independent of the order of its factors (Transfer homomorphism for a finite index subgroup).

[F2]

If $u, v \in G$ represent the same right coset of $H$ , that is $H u = H v$ , then $u v^{- 1} \in H$ ; conversely $u v^{- 1} \in H$ implies $H u = H v$ ( $x \in a H$ iff $a^{- 1} x \in H$ , and $a H = b H$ iff $a^{- 1} b \in H$ , Left and right cosets $g H$ and $H g$ of a subgroup).

[F3]

For all $u, v \in H$ the homomorphism $φ$ satisfies $φ (u v) = φ (u) φ (v)$ , $φ (e) = 1$ and $φ (u^{- 1}) = φ (u)^{- 1}$ (A group homomorphism automatically satisfies $f (e) = e^{'}$ and $f (g^{- 1}) = f (g)^{- 1}$ , and $f (g^{n}) = f (g)^{n}$ for every $n \in Z$ ; for monoid homomorphisms preservation of the identity must be assumed, Monoid homomorphism and group homomorphism).

[F4]

The map $α \mapsto α x$ is a bijection of the finite set $H \ G$ , so a product indexed by $H \ G$ may be reindexed along it (The coset set $G / H$ and the index $[G : H]$ of a subgroup, Transfer homomorphism for a finite index subgroup).

Proof

technique · direct

1.1

For every $α \in H \ G$ one has $t_{α}^{'} = h_{α} t_{α}$ for some $h_{α} \in H$ : both $t_{α}^{'}$ and $t_{α}$ represent the coset $α$ , so $t_{α}^{'} t_{α}^{- 1} \in H$ by [F2], and $h_{α} : = t_{α}^{'} t_{α}^{- 1}$ is the desired element.

F2given

2.1

For $x \in G$ and $α \in H \ G$ the product $t_{α}^{'} x t_{α x}^{'}^{- 1}$ equals $h_{α} t_{α} x t_{α x}^{- 1} h_{α x}^{- 1}$ , because $t_{α}^{'} = h_{α} t_{α}$ and $t_{α x}^{'} = h_{α x} t_{α x}$ by step 1.1; hence $φ (t_{α}^{'} x t_{α x}^{'}^{- 1}) = φ (h_{α}) φ (t_{α} x t_{α x}^{- 1}) φ (h_{α x})^{- 1}$ by [F3].

F3step 1.1algebra

2.2

The reindexing $α \mapsto α x$ is a bijection of $H \ G$ by [F4], so $\prod_{α} φ (h_{α x})^{- 1} = \prod_{β} φ (h_{β})^{- 1} = (\prod_{β} φ (h_{β}))^{- 1}$ by [F3].

F3F4step 1.1

3.1

Consequently $V^{'} (x) = (\prod_{α} φ (h_{α})) (\prod_{α} φ (t_{α} x t_{α x}^{- 1})) (\prod_{α} φ (h_{α x})^{- 1})$ : the product over $α$ of the three factors of step 2.1 may be rearranged because $A$ is abelian, by [F1].

F1step 2.1

4.1

Therefore $V^{'} (x) = (\prod_{α} φ (h_{α})) V (x) (\prod_{α} φ (h_{α}))^{- 1} = V (x)$ , the two outer factors cancelling because multiplication in the abelian group $A$ commutes.

F1step 3.1step 2.2algebra

5.1

Since $x \in G$ was arbitrary, $V^{'} = V$ ; the transfer is therefore independent of the choice of transversal. ∎

step 4.1given

Depends on

Used by

Dependency tree · two levels

26 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Paul Flavell, An Introduction to Transfer and Fusion in Finite Groups, §§2–5 (standard reference, not scraped)
Hans Kurzweil and Bernd Stellmacher, The Theory of Finite Groups, §§7.1–7.2 (standard reference, not scraped)