The Commuting Algebra of an Involution

Introduction

An involutive automorphism $\alpha$ of a group $G$ has a fixed subgroup $G^{\alpha}$, and it also has a larger set of elements with which it is compatible in a weaker sense: those whose inner automorphism commutes with $\alpha$. That set is a subgroup, it contains the fixed subgroup and the centre, and it is exactly the set of elements whose conjugacy defect under $\alpha$ is central. It is the group-level form of the centraliser of the involution, and it is what this article calls the commuting algebra of the involution; the name is justified at the end, where the group algebra is mentioned, and the group itself is the object throughout. The article is the fourth of the involutive *-articles; the fixed subgroup is owned by Involutions and the Fixed-Point Subgroup, the semidirect product by Involutive Groups, §9, and the inner automorphism group by The Conjugation Representation.

Throughout, $(G,\sigma)$ is an involutive group, $\alpha=\sigma\iota$ is the associated involutive automorphism, $\iota$ is the inversion, $c_g$ is the inner automorphism $x\mapsto gxg^{-1}$, and the action of a subgroup on the fixed subgroup follows Involutive Group Actions.

The Commuting Subgroup

Definition. An element $g\in G$ commutes with the involutive automorphism $\alpha$ if the inner automorphism $c_g$ commutes with $\alpha$ as a map, $c_g\alpha=\alpha c_g$. The commuting set is

$$ C_G(\alpha)=\{g\in G : c_g\alpha=\alpha c_g\}. $$

Proposition (the commuting condition is centrality of the defect). For $g\in G$,

$$ c_g\alpha=\alpha c_g \quad\Longleftrightarrow\quad g\,\alpha(g)^{-1}\in Z(G) \quad\Longleftrightarrow\quad \alpha(g)\,g^{-1}\in Z(G). $$

Proof. Writing the composition out, $c_g\alpha(x)=g\alpha(x)g^{-1}$ and $\alpha c_g(x)=\alpha(gxg^{-1})=\alpha(g)\alpha(x)\alpha(g)^{-1}$ for all $x$; the two are equal for all $x$ exactly when $g^{-1}\alpha(g)$ centralises $\alpha(G)=G$, that is $g^{-1}\alpha(g)\in Z(G)$. The second form is the inverse of the first.

Theorem (the commuting set is a subgroup). $C_G(\alpha)$ is a subgroup of $G$ containing $Z(G)$ and $G^{\alpha}$.

Proof. The identity commutes with $\alpha$. If $g\in C_G(\alpha)$ then $\alpha(g^{-1})=\alpha(g)^{-1}=(z g)^{-1}=g^{-1}z^{-1}$ for some $z\in Z(G)$, so $g^{-1}\alpha(g^{-1})^{-1}=g^{-1}\alpha(g)=g^{-1}zg\in Z(G)$, and $g^{-1}\in C_G(\alpha)$. If $g,h\in C_G(\alpha)$, write $\alpha(g)=z_1g$ and $\alpha(h)=z_2h$ with $z_1,z_2\in Z(G)$; then $\alpha(gh)=z_1z_2\,gh$, so $(gh)\alpha(gh)^{-1}=(z_1z_2)^{-1}\in Z(G)$ and $gh\in C_G(\alpha)$. The centre is contained because a central $g$ has $c_g=\mathrm{id}$, which commutes with everything; the fixed subgroup is contained because $\alpha(g)=g$ gives $g\alpha(g)^{-1}=e$.

Proposition (the form of $\alpha$ on the commuting subgroup). The restriction $\alpha|_{C_G(\alpha)}$ multiplies by a central element: the map

$$ z : C_G(\alpha)\longrightarrow Z(G), \qquad z(g)=\alpha(g)\,g^{-1}, $$

is a homomorphism whose kernel is the fixed subgroup $G^{\alpha}$, and $\alpha(g)=z(g)g$ for every $g\in C_G(\alpha)$.

Proof. The containment in $Z(G)$ is the proposition above. The multiplicativity follows from the central value: $z(gh)=\alpha(gh)(gh)^{-1}=\alpha(g)\alpha(h)h^{-1}g^{-1}=\alpha(g)z(h)g^{-1}=z(h)\alpha(g)g^{-1}=z(h)z(g)$, using that $z(h)$ is central. The kernel is $\{g:\alpha(g)g^{-1}=e\}=\{g:\alpha(g)=g\}=G^{\alpha}$.

Corollary (the exact sequence). The homomorphism $z$ realises the commuting subgroup as an extension of the fixed subgroup by a subgroup of the centre, and the image of $G\to\operatorname{Inn}(G)$ on $C_G(\alpha)$ is exactly the centraliser $C_{\operatorname{Inn}(G)}(\alpha)$ of $\alpha$ in the inner automorphism group. Consequently

$$ C_G(\alpha)/Z(G)\ \cong\ C_{\operatorname{Inn}(G)}(\alpha), \qquad\text{and}\qquad C_G(\alpha)/G^{\alpha}\ \cong\ \operatorname{im}z\ \leq Z(G). $$

Proof. The first isomorphism is that the kernel of $G\to\operatorname{Inn}(G)$ is $Z(G)$ and the image on $C_G(\alpha)$ is the set of inner automorphisms commuting with $\alpha$. The second is the first isomorphism theorem applied to $z$.

Relation to the Fixed Subgroup

Proposition (the containment and its failure to be an equality). $G^{\alpha}\subseteq C_G(\alpha)$, and the inclusion is in general strict. It is an equality exactly when every element commuting with $\alpha$ is fixed by it, that is when $\alpha$ has no central defect on $C_G(\alpha)$.

Proof. The containment is the theorem. For the strictness, in the dihedral group $D_4=\langle r,s\mid r^{4}=s^{2}=e,\ srs=r^{-1}\rangle$ take $\alpha=c_s$, the conjugation by the reflection $s$; then $G^{\alpha}=C_{D_4}(s)=\{e,r^{2},s,sr^{2}\}$ has four elements, while $r\notin G^{\alpha}$ yet $r\alpha(r)^{-1}=r\cdot r=r^{2}\in Z(D_4)=\{e,r^{2}\}$, so $r\in C_G(\alpha)$ and the containment is strict.

Proposition (the commuting subgroup of the identity and of the inversion). If $\alpha=\mathrm{id}$ then $C_G(\alpha)=G$; if $\alpha=\iota$ then $C_G(\alpha)=\{g:g^{2}\in Z(G)\}$, the elements whose square is central.

Proof. For the identity the defect is $g\alpha(g)^{-1}=e$ for every $g$. For the inversion, $\alpha(g)g^{-1}=g^{-1}g^{-1}=g^{-2}$, which is central exactly when $g^{2}$ is central.

Corollary (two degenerate cases). On an abelian group the commuting subgroup of the inversion is the whole group; on a group of exponent two the commuting subgroup of every involutive automorphism is the whole group.

Proof. On an abelian group every square is central, so the criterion of the proposition holds for every element. On a group of exponent two every element satisfies $g^{2}=e$, hence $g=g^{-1}$, and for an involutive automorphism $\alpha$ one has $\alpha(g)=g^{-1}=g$; the defect $g\alpha(g)^{-1}$ is $e$ for every $g$, so $C_G(\alpha)=G$.

The Commuting Subgroup and the Semidirect Product

Let $t$ be the element of order two acting on $G$ by $\alpha$ in the semidirect product $G\rtimes\langle t\rangle$ of Involutive Groups, §9, so that $tgt^{-1}=\alpha(g)$.

Proposition (the commutation with $t$). For $g\in G$ the commutator with $t$ is $[g,t]=g\alpha(g)^{-1}$, and

$$ C_G(\alpha)=\{g\in G : [g,t]\in Z(G)\}. $$

Proof. $[g,t]=gtg^{-1}t^{-1}=g\,\alpha(g)^{-1}\,t t^{-1}=g\alpha(g)^{-1}$, using $t^{-1}=t$ and $tg^{-1}t^{-1}=\alpha(g^{-1})=\alpha(g)^{-1}$. The description of $C_G(\alpha)$ is then the proposition above.

Proposition (the centraliser of $t$). In $G\rtimes\langle t\rangle$ the centraliser of $t$ meets $G$ in $G^{\alpha}$: $C_{G\rtimes\langle t\rangle}(t)\cap G=G^{\alpha}$. Hence the elements commuting with $\alpha$ are the elements whose commutator with $t$ is central, and the elements fixed by $\alpha$ are the elements commuting with $t$.

Proof. $g t=t g$ is $tgt^{-1}=g$, that is $\alpha(g)=g$; so $C_{G\rtimes\langle t\rangle}(t)\cap G=G^{\alpha}$. The second statement restates the description of $C_G(\alpha)$.

Corollary (the commuting subgroup as a preimage). $C_G(\alpha)$ is the largest subgroup $H$ of $G$ containing $Z(G)$ with $[H,t]\subseteq Z(G)$; it is the full preimage under $G\to G/Z(G)$ of the centraliser of the coset $tZ(G)$ in $(G\rtimes\langle t\rangle)/Z(G)$.

Proof. The first statement is the definition read through the commutator description and the closure proved in the theorem. The second is the exact sequence, since the image of $G$ in $(G\rtimes\langle t\rangle)/Z(G)$ centralises the coset of $t$ exactly when $[g,t]\in Z(G)$.

The Name and the Group Algebra

Remark (why "algebra"). In the group $G$ the object defined here is a subgroup, and the word "algebra" in the title refers to its linearisation. The involutive automorphism $\alpha$ extends linearly to an automorphism of the group algebra $k[G]$, the fixed set $k[G]^{\alpha}=\{a:\alpha(a)=a\}$ is a subalgebra of $k[G]$, and the commuting subgroup is the preimage of the centraliser of $\alpha$ in the inner automorphism group computed in the previous section. The full treatment of $k[G]$, of the fixed subalgebra and of the skew group algebra belongs to Group Algebras and to the representation theory of later parts; nothing of it is used here.

Remark (the term for the working reader). The name "commuting algebra" is used here with the reading fixed by the scope of the article: the elements of the group commuting with the involutive automorphism in the sense that their inner automorphisms commute with it. A sibling reading of the same name, the fixed subalgebra of the group algebra, is the second object of this section and is cited, not developed.

Summary

For an involutive automorphism $\alpha$ of $G$ the commuting set $C_G(\alpha)=\{g\in G : c_g\alpha=\alpha c_g\}$ consists of the elements whose conjugate defect $g\alpha(g)^{-1}$ is central, and it is a subgroup containing $Z(G)$ and the fixed subgroup $G^{\alpha}$. On it, $\alpha$ multiplies by the central element $z(g)=\alpha(g)g^{-1}$, the map $z$ is a homomorphism $C_G(\alpha)\to Z(G)$ with kernel $G^{\alpha}$, and

$$ C_G(\alpha)/Z(G)\cong C_{\operatorname{Inn}(G)}(\alpha), \qquad C_G(\alpha)/G^{\alpha}\cong\operatorname{im}z\leq Z(G). $$

The fixed subgroup is contained and the containment is strict in general, the dihedral group $D_4$ with the conjugation by a reflection giving $G^{\alpha}$ of order four inside $C_G(\alpha)=G$ of order eight. For $\alpha=\mathrm{id}$ the commuting set is all of $G$, and for $\alpha=\iota$ it is the set of elements whose square is central.

In the semidirect product $G\rtimes\langle t\rangle$ with $t$ acting by $\alpha$, the commutator is $[g,t]=g\alpha(g)^{-1}$, so $C_G(\alpha)=\{g:[g,t]\in Z(G)\}$, the centraliser of $t$ meets $G$ in $G^{\alpha}$, and $C_G(\alpha)$ is the full preimage of the centraliser of the coset of $t$ in the quotient by the centre. The name "commuting algebra" refers to the linearisation: $\alpha$ extends to the group algebra $k[G]$, whose fixed subalgebra is cited to Group Algebras.

Summary of Notation

Symbol	Meaning
$c_g\alpha=\alpha c_g$	$g$ commutes with the involutive automorphism
$C_G(\alpha)$	commuting subgroup, the commuting algebra of $\alpha$
$g\alpha(g)^{-1}\in Z(G)$	centrality of the conjugate defect
$z(g)=\alpha(g)g^{-1}$	homomorphism $C_G(\alpha)\to Z(G)$, kernel $G^{\alpha}$
$C_G(\alpha)/Z(G)\cong C_{\operatorname{Inn}(G)}(\alpha)$	the exact sequence
$[g,t]=g\alpha(g)^{-1}$	commutator in the semidirect product
$k[G]^{\alpha}$	fixed subalgebra of the group algebra, the linearisation