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Transfer cycle decomposition formula

Statement

Let $G$ be a finite group, $H \leq G$ , $A$ an abelian group written multiplicatively, $φ : H \to A$ a homomorphism, and $V_{φ} : G \to A$ the transfer of Transfer homomorphism for a finite index subgroup, which is independent of the transversal by Transfer is independent of the transversal. Let $x \in G$ and let

$H \ G = C_{1} ⊔ \dots ⊔ C_{r}$

be the decomposition of the finite set of right cosets into the orbits of the right multiplication action of the cyclic subgroup $⟨ x ⟩ \leq G$ (The subgroup $⟨ S ⟩$ generated by a subset, the cyclic subgroup $⟨ g ⟩$ , and cyclic groups, $⟨ g ⟩ = {g^{n} : n \in Z}$ , and every cyclic group is abelian). Put $n_{i} : = ∣ C_{i} ∣$ and choose $t_{i} \in G$ with $H t_{i} \in C_{i}$ . Then

$V_{φ} (x) = \prod_{i = 1}^{r} φ (t_{i} x^{n_{i}} t_{i}^{- 1}) .$

Facts & Assumptions

Given: A finite group $G$ , a subgroup $H \leq G$ , an abelian group $A$ , a homomorphism $φ : H \to A$ , an element $x \in G$ , and the transfer $V = V_{φ}$ of Transfer homomorphism for a finite index subgroup.

[F1]

For $α \in H \ G$ the coset $α t = H t_{α} t$ is defined for every $t \in G$ , the assignment $α \mapsto α t$ is a right action of $G$ on the finite set $H \ G$ , each factor $t_{α} s t_{α s}^{- 1}$ ( $s \in G$ ) lies in $H$ , and products of finitely many elements of the abelian group $A$ are independent of the order of the factors (Transfer homomorphism for a finite index subgroup).

[F2]

The transfer $V (x) = \prod_{α \in H \ G} φ (t_{α} x t_{α x}^{- 1})$ does not depend on the transversal (Transfer is independent of the transversal).

[F3]

If $u, v \in G$ satisfy $H u = H v$ then $u v^{- 1} \in H$ , and conversely ( $x \in a H$ iff $a^{- 1} x \in H$ , and $a H = b H$ iff $a^{- 1} b \in H$ ).

[F4]

$φ (u v) = φ (u) φ (v)$ and $φ (e) = 1$ for $u, v \in H$ (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).

[F5]

$⟨ x ⟩ = {x^{k} : k \in Z}$ is a subgroup of $G$ , and $x^{j} x^{k} = x^{j + k}$ for all $j, k \in Z$ ( $⟨ g ⟩ = {g^{n} : n \in Z}$ , and every cyclic group is abelian, Exponent laws in a group: $g^{m + n} = g^{m} g^{n}$ and $(g^{m})^{n} = g^{m n}$ for all $m, n \in Z$ , and $(g h)^{n} = g^{n} h^{n}$ when $g$ and $h$ commute).

[F6]

The orbits of the action of the subgroup $⟨ x ⟩$ on $H \ G$ partition $H \ G$ (The orbits of a group action are the equivalence classes of $x \sim y$ iff $y = g \cdot x$ for some $g$ , and hence partition the acted-on set, The orbit $G \cdot x$ and stabilizer $G_{x}$ of a point in a group action, Left group actions, transitive actions, and faithful actions).

[F7]

Every nonempty set of positive integers has a least element, and for integers $k$ and $n \geq 1$ there are $q, r \in Z$ with $k = q n + r$ and $0 \leq r < n$ (The well-ordering principle, Division with remainder in $Z$ : for $a \in Z$ and $b > 0$ there are unique $q, r \in Z$ with $a = q b + r$ and $0 \leq r < b$ ).

Proof

technique · direct

1.1

Fix $α \in H \ G$ . Since $H \ G$ is finite, the elements $α, α x, α x^{2}, \dots$ cannot all be distinct, so $α x^{i} = α x^{j}$ for some $0 \leq i < j$ ; applying the inverse permutation $α \mapsto α x^{- i}$ (which is the action of $x^{- i}$ ) gives $α x^{j - i} = α$ , an equality of the form $α x^{n} = α$ with $n = j - i \geq 1$ . By [F7] there is a least positive integer $n (α)$ with $α x^{n (α)} = α$ .

F1F5F7algebra

2.1

For every $k \in Z$ one has $α x^{k} = α$ if and only if $n (α) ∣ k$ : if $k = q n (α)$ then $α x^{k} = α$ by [F5] and step 1.1 (including $q < 0$ , since $α x^{- n (α)} = α$ as well), while for arbitrary $k$ writing $k = q n (α) + r$ with $0 \leq r < n (α)$ by [F7] gives $α x^{r} = α$ , so minimality forces $r = 0$ .

F5F7step 1.1

3.1

The orbit of $α$ under $⟨ x ⟩$ is $C (α) = {α x^{k} : k \in Z}$ , and by step 2.1 it equals ${α, α x, \dots, α x^{n (α) - 1}}$ , whose $n (α)$ elements are pairwise distinct. In particular $α x^{n (α)} = α$ and $∣ C (α) ∣ = n (α)$ .

step 2.1

4.1

The orbit decomposition $H \ G = C_{1} ⊔ \dots ⊔ C_{r}$ of [F6] is therefore a decomposition into finitely many sets $C_{i}$ each of the form $C_{i} = {α_{i}, α_{i} x, \dots, α_{i} x^{n_{i} - 1}}$ with $n_{i} = ∣ C_{i} ∣$ and $α_{i} x^{n_{i}} = α_{i}$ ; write $α_{i} = H t_{i}$ .

F6step 3.1given

5.1

The rule $t_{α_{i} x^{j}} : = t_{i} x^{j}$ for $0 \leq j < n_{i}$ defines a transversal of $H \ G$ : indeed every coset of $H \ G$ lies in exactly one $C_{i}$ and hence equals exactly one $α_{i} x^{j}$ , and $t_{i} x^{j}$ represents it, because $H t_{i} x^{j} = α_{i} x^{j}$ by [F1].

F1step 4.1

6.1

Because $V$ is independent of the transversal by [F2], it may be computed with the transversal of step 5.1: $V (x) = \prod_{i = 1}^{r} \prod_{j = 0}^{n_{i} - 1} φ (t_{α_{i} x^{j}} x t_{(α_{i} x^{j}) x}^{- 1})$ , where $α : = α_{i} x^{j}$ and $α x = α_{i} x^{j + 1}$ .

F1F2step 5.1

7.1

For $0 \leq j < n_{i} - 1$ one has $t_{(α_{i} x^{j}) x} = t_{α_{i} x^{j + 1}} = t_{i} x^{j + 1} = t_{i} x^{j} x = t_{α_{i} x^{j}} x$ , so the corresponding factor equals $φ (t_{α_{i} x^{j}} x (t_{α_{i} x^{j}} x)^{- 1}) = φ (e) = 1$ by [F4].

F4step 6.1algebra

7.2

For $j = n_{i} - 1$ one has $α_{i} x^{j + 1} = α_{i} x^{n_{i}} = α_{i}$ by step 4.1, so $t_{(α_{i} x^{n_{i} - 1}) x} = t_{α_{i}} = t_{i}$ and the corresponding factor equals $φ (t_{i} x^{n_{i} - 1} x t_{i}^{- 1}) = φ (t_{i} x^{n_{i}} t_{i}^{- 1})$ , an element of $H$ because $H t_{i} x^{n_{i}} = H t_{i}$ by step 4.1 and [F3].

F3step 4.1step 6.1algebra

8.1

Multiplying the contributions of steps 7.1 and 7.2 over all $i$ and $j$ , all factors with $j < n_{i} - 1$ are $1$ and the remaining one for each $i$ is $φ (t_{i} x^{n_{i}} t_{i}^{- 1})$ ; hence $V (x) = \prod_{i = 1}^{r} φ (t_{i} x^{n_{i}} t_{i}^{- 1})$ . ∎

step 7.1step 7.2given

Depends on

Used by

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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)