Free Involutions and the Quotient

Introduction

An involution of a group always fixes the identity, and it is free when it fixes nothing else. A free involution generates an action of the two-element group on the underlying set whose orbits are all of size two, so the orbit space is a quotient over which the involution is invisible, and the projection is a two-to-one map. This article fixes the notion, constructs the quotient and the double cover, and treats the two standard instances: the inversion on a group with no element of order two, whose extension is dihedral, and the negation on the additive group of a field of characteristic not two. It is the third of the involutive *-articles; the involution, its fixed set, the associated involutive automorphism and the split extension are owned by Involutive Groups, and the action of a group on an involutive group by Involutive Group Actions.

Throughout, $(G,\sigma)$ is an involutive group with $\sigma$ an anti-automorphism of order two, $\alpha=\sigma\iota$ is the associated involutive automorphism, $G^{\sigma}=\{g:\sigma(g)=g\}$ and $I(\sigma)=\{g:\sigma(g)=g^{-1}\}$ are the fixed and inverted sets, and $\iota$ is the inversion.

Free Involutions and Their Dual

Definition. The involution $\sigma$ is free, or fixed-point-free, if $G^{\sigma}=\{e\}$. The involutive automorphism $\alpha$ is free if $G^{\alpha}=\{e\}$.

Proposition (the two fixed sets in dual form). $G^{\sigma}=I(\alpha)$ and $G^{\alpha}=I(\sigma)$. Hence $\sigma$ is free if and only if $I(\alpha)=\{e\}$, and $\alpha$ is free if and only if $I(\sigma)=\{e\}$.

Proof. $g\in G^{\sigma}$ means $\sigma(g)=g$; applying $\iota$ and using $\alpha=\sigma\iota$ gives $\alpha(g)=g^{-1}$, that is $g\in I(\alpha)$. The second statement is the same with the roles of $\sigma$ and $\alpha$ exchanged, and the versions for freeness follow. Both identities are recorded in Involutive Groups, §4.

Remark (the two notions are dual and not equivalent). Freeness of $\sigma$ and freeness of $\alpha$ are different conditions, and neither implies the other. On the cyclic group $C_3=\langle r\rangle$ the inversion $\sigma=\iota$, which is an automorphism because $C_3$ is abelian, is free: $\sigma(r)=r^{-1}\neq r$ and $\sigma(r^{2})=r\neq r^{2}$. Its associated automorphism is $\alpha=\sigma\iota=\mathrm{id}$, which fixes every element and is as far from free as possible. So a free involution may have an associated automorphism that is not free. The asymmetry is exactly the asymmetry between the fixed and the inverted sets.

Proposition (the canonical free involution). The inversion $\iota$ is free if and only if $G$ has no element of order two; for a finite group this is the condition that $\lvert G\rvert$ is odd.

Proof. $\iota(g)=g$ is $g^{-1}=g$, that is $g^{2}=e$; so the fixed set of the inversion is the $2$-torsion of $G$, which is $\{e\}$ exactly when no element of order two exists. A finite group has an element of order two exactly when its order is even (Cauchy's theorem).

Proposition (the injectivity of the conjugate product). If $\sigma$ is free, the map

$$ \varphi : G\longrightarrow G, \qquad \varphi(g)=g^{-1}\sigma(g), $$

is injective. If $G$ is finite it is therefore a bijection, and every element of $G$ is the conjugate product of a unique pair $(g^{-1},\sigma(g))$.

Proof. If $\varphi(g)=\varphi(h)$ then $g^{-1}\sigma(g)=h^{-1}\sigma(h)$, so $hg^{-1}=\sigma(h)\sigma(g)^{-1}=\sigma(hg^{-1})$, and $hg^{-1}\in G^{\sigma}$; freeness gives $hg^{-1}=e$ and $g=h$. Finiteness makes injective bijective.

The Quotient and the Double Cover

Definition. The orbit quotient of a free involution $\sigma$ is the set of orbits

$$ G/\langle\sigma\rangle = \{\,\{g,\sigma(g)\} : g\in G\,\} , \qquad \pi : G\longrightarrow G/\langle\sigma\rangle, \quad \pi(g)=\{g,\sigma(g)\}. $$

Proposition (the projection is a double cover). The projection $\pi$ is surjective, every fibre has exactly two elements, and $\sigma$ is its nontrivial deck transformation: $\pi\sigma=\pi$ and $\sigma(g)\neq g$ for $g\neq e$. For finite $G$, $\lvert G/\langle\sigma\rangle\rvert=\lvert G\rvert/2$.

Proof. The fibre over $\{g,\sigma(g)\}$ is that orbit, of size two because $\sigma(g)=g$ forces $g=e$ and an orbit of a non-identity element has two members. And $\pi(\sigma(g))=\{\sigma(g),\sigma^{2}(g)\}=\{g,\sigma(g)\}=\pi(g)$. The count is the orbit-counting of a free action of a two-element group (Transformation Groups).

Proposition (the quotient is not a group in the natural way). For $G\neq\{e\}$ the quotient $G/\langle\sigma\rangle$ carries no group structure for which $\pi$ is a homomorphism.

Proof. A homomorphism with trivial kernel is injective: if $\pi(g)=\pi(h)$ then $\pi(gh^{-1})=\pi(g)\pi(h)^{-1}=e$, so $gh^{-1}\in\ker\pi=\{e\}$ and $g=h$. But $\pi$ is two-to-one on the non-identity elements, so it is not injective; no such group structure exists.

Definition. The quotient is the base and $\pi$ the double cover. The group-theoretic double cover attached to the involutive automorphism $\alpha$ is the split extension

$$ 1\longrightarrow G\longrightarrow G\rtimes\langle t\rangle\longrightarrow \langle t\rangle\longrightarrow 1, \qquad t^{2}=e, $$

of Involutive Groups, §9, a group of order $2\lvert G\rvert$ in which $G$ is normal of index two and conjugation by $t$ realises $\alpha$.

Remark (which involution is free). The set-theoretic quotient and double cover are built from the involution $\sigma$ and only need it to be an involution. The group-theoretic extension is built from the involutive automorphism $\alpha$, which is the datum that acts on $G$ by automorphisms; the two agree when $\sigma$ is itself an automorphism, that is when $G$ is abelian.

The Finite Case

Theorem. A finite abelian group admitting a free involutive automorphism is of odd order, and the automorphism is the inversion. Hence its group-theoretic double cover is the dihedral group of the group.

Proof. Let $G$ be finite abelian and $\alpha$ a free involutive automorphism. The map $\psi(g)=\alpha(g)g^{-1}$ is an endomorphism of $G$, and its kernel is $G^{\alpha}=\{e\}$; being injective on a finite group, it is bijective. Now

$$ \psi(\alpha(g))=\alpha^{2}(g)\alpha(g)^{-1}=g\,\alpha(g)^{-1}=\bigl(\alpha(g)g^{-1}\bigr)^{-1}=\psi(g)^{-1}=\psi(g^{-1}), $$

so $\psi(\alpha(g))=\psi(g^{-1})$; injectivity of $\psi$ gives $\alpha(g)=g^{-1}$ for every $g$. Finally $\alpha(g)=g^{-1}$ has fixed set the $2$-torsion, which is $\{e\}$ exactly when the order is odd.

Corollary (the general finite statement). A finite group admitting a free involutive automorphism is abelian of odd order with the inversion as the automorphism. The abelian case is proved above; the general case is a theorem of the finite-group literature and is stated here without proof.

The Standard Examples

Example (the dihedral example). Let $G=C_n=\langle r\rangle$ with $n$ odd and let $\sigma=\iota$ be the inversion, which is an automorphism because $C_n$ is abelian and is free because $n$ is odd. The orbits are the pairs $\{r^{k},r^{-k}\}$, the quotient has $(n+1)/2$ elements, and the group-theoretic double cover is the dihedral group $D_n=C_n\rtimes C_2$ of Involutive Groups, §9, in which the complement acts by the inversion $r\mapsto r^{-1}$. In $D_n$ the rotations are the paired elements and the reflections are the added coset; the double cover $C_n\to C_n/\langle\iota\rangle$ is the shadow of the split extension on the underlying sets.

Example (the antipodal example). Let $F$ be a field of characteristic different from two and let $G=(F,+)$ be its additive group with $\sigma(x)=-x$. The involution is free, since $-x=x$ is $2x=0$ and $2$ is invertible in $F$; the orbits are the antipodal pairs $\{x,-x\}$, and the quotient has one point for each pair. The group-theoretic double cover is the semidirect product $F\rtimes C_2$ in which the complement acts by negation. Over a field of characteristic two the negation is the identity, which is not free, so freeness is a property of the pair (group, field) and not of the additive group alone.

Example (the infinite case). Let $G=(\mathbb{Z},+)$ with $\sigma(n)=-n$. The involution is free, the quotient has countably many pairs, and the extension is the infinite dihedral group $\mathbb{Z}\rtimes C_2$. The conjugate-product map of the first section is $\varphi(n)=-2n$ in additive notation, a bijection of $\mathbb{Z}$; the finite theorem does not apply, but the quotient and the double cover are constructed in the same way.

Remark (the two examples in one). The dihedral and antipodal examples are the same construction read on two free involutions: the inversion of a cyclic group of odd order, and the negation of the additive group of a field of characteristic different from two. Both are free, both give a two-to-one quotient, and both extend to a semidirect product by the two-element group; they differ in the size and in the nature of the group, one finite and cyclic, the other the additive group of a field.

Summary

An involution $\sigma$ of $G$ is free when $G^{\sigma}=\{e\}$, and the associated automorphism $\alpha=\sigma\iota$ is free when $G^{\alpha}=\{e\}$; by the dual identities $G^{\sigma}=I(\alpha)$ and $G^{\alpha}=I(\sigma)$ the two notions are the vanishing of the two inverted sets, and neither implies the other, as $C_3$ with its free inversion and its identity automorphism shows. The canonical free involution is the inversion on a group with no element of order two; on a finite group this is odd order.

A free involution generates a free action of the two-element group on the underlying set, so every orbit has two elements and the orbit quotient $G/\langle\sigma\rangle$ is a set with a two-to-one projection $\pi$, the double cover, whose nontrivial deck transformation is $\sigma$ and whose cardinality is $\lvert G\rvert/2$ for finite $G$. The projection carries no group structure, because a homomorphism with trivial kernel is injective and $\pi$ is two-to-one. The group-theoretic double cover uses the involutive automorphism $\alpha$ and is the split extension $G\rtimes C_2$ of Involutive Groups, §9.

A finite abelian group with a free involutive automorphism has odd order and the automorphism is the inversion, the abelian case being proved by the bijectivity of $g\mapsto\alpha(g)g^{-1}$; the general finite case, without the abelian hypothesis, is a theorem of the literature. The standard free involutions are the inversion of a cyclic group of odd order, whose extension is the dihedral group, and the negation of the additive group of a field of characteristic different from two, the antipodal involution.

Summary of Notation

Symbol	Meaning
free, $G^{\sigma}=\{e\}$	the involution fixes only the identity
$G^{\sigma}=I(\alpha)$	dual identity for the fixed and inverted sets
$\varphi(g)=g^{-1}\sigma(g)$	injective map of a free involution
$G/\langle\sigma\rangle$	orbit quotient, the base of the double cover
$\pi(g)=\{g,\sigma(g)\}$	two-to-one projection, the double cover
$\lvert G/\langle\sigma\rangle\rvert=\lvert G\rvert/2$	orbit count for a finite free involution
$G\rtimes C_2$	group-theoretic double cover, the split extension
$\psi(g)=\alpha(g)g^{-1}$	bijective endomorphism proving $\alpha=\iota$ in the abelian case