nLab centralizer

Definition

Definition

Given a group $G$ and a subset $S \,\subset\, G$ of its underlying set, the centralizer subgroup (also: the commutant) of $S$ in $G$ is the subgroup

$$C_G(S) \;\coloneqq\; \big\{ g \in G \,\vert\, \underset{s \in S}{\forall} ( g \cdot s \,=\, s \cdot g ) \big\} \;\subset\; G$$

of all elements $c \in G$ which commute with the elements of $S$.

Equivalently, the centralizer is the joint fixed point subgroup of the inner automorphisms on $G$ given by conjugation with the elements $s \in S$.

Notice the similarity with but the difference to the concept of normalizer subgroup, cf. Prop. .

Properties

Proposition

Given a subset $S \subset G$ of a group $G$, the centralizer subgroup of $S$ (Def. ) is a subgroup of the normalizer subgroup:

$$C_G(S) \; \subset \; N_G(S) \,.$$

Proof

Since an element $g \in G$ which fixes each element $s \in S$ separately already fixes the entire subset as such:

$$\underset{s \in S}{\forall} \big( g \cdot s \,=\, s \cdot g \big) \;\;\;\;\; \Rightarrow \;\;\;\;\; \big( g \cdot S \,=\, S \cdot g \big) \,.$$

Proposition

(centralizers in $T_1$-groups are closed)
If $G$ is a $T_1$-topological group, then all its centralizer subgroups are closed subgroups.

Proof

First consider a singleton set $S = \{s\}$. By definition, the centralizer of a single element $s \in G$ is the preimage of itself under the function

$$\array{ G &\xrightarrow{\;\;}& G \\ g &\mapsto& g \cdot s \cdot g^{-1} \,. }$$

(the adjoint action of $G$ on itself).

Noticing here that:

1. this is continuous function, by the axioms on a topological group;

2. $\{s\} \subset G$ is a closed subset, by the assumption that $G$ is a $T_1$-space (by this Prop.)

it follows that $C_G(\{s\}) \subset G$ is the continuous preimage of a closed subset and hence is itself closed (by this Prop.).

Now for a general set $S$, its centralizer is clearly the intersection of the centralizers of (the singleton sets on) its elements:

$$C_G(S) \;=\; \underset{ s \in S }{\cap} C_G\big(\{s\}\big) \,.$$

Since each of the factors on the right isclosed, by the previous argument, the general centralizer subgroup is an intersection of closed subsets and hence itself a closed subset.

Examples

In homotopy long exact sequences

Consider a path-connected topological space $X$ admitting the structure of a CW-complex.

Fixing any base points $0 \in X$ and $0 \in S^1$, we will be concerned with the free loop space

$$\mathcal{L} X \coloneqq \mathrm{Map}({ S^1, X })$$

and the based loop space

$$\Omega X \coloneqq \mathrm{Map}^\ast({ S^1, X }) \mathrlap{\,.}$$

Their sets of connected components are the fundamental group of $X$

$$G \coloneqq \pi_1(X) \equiv \pi_0({\Omega X})$$

and its set of conjugacy classes:

$$\mathrm{Conj}(G) \simeq \pi_0({\mathcal{L}X}) \mathrlap{\,.}$$

Proposition

In every connected component $[g] \in \mathrm{Conj}(G) \simeq \pi_0(\mathcal{L}X)$, the image of $\pi_1({\mathcal{L}X})$ in $G$ is the centralizer group of $g$:

(1)$$\pi_1(\mathrm{ev}) \big({ \pi_1({ \mathcal{L}X }) }\big) \simeq C_G(g) \subset G \mathrlap{\,.}$$

Moreover, when $X$ has trivial $\pi_2$, then $\pi_1({\mathcal{L}X})$ is isomorphic to the centralizer

(2)$$\mathllap{ \pi_2(X) \simeq \ast \;\;\;\;\; \Rightarrow \;\;\;\;\; } \pi_1({ \mathcal{L}X }) \simeq C_G(g) \mathrlap{\,.}$$

Proof

Consider any loop $\gamma \in \Omega X$ which represents the conjugacy class $[g]$:

and with that used as the base point, consider the homotopy long exact sequence induced by the map $\mathrm{ev}$ that evaluates a loop at its base point:

$$\Omega X \longrightarrow \mathcal{L}X \xrightarrow{\phantom{-} ev \phantom{-}} X \mathrlap{\,,}$$

of this form:

On the right we are claiming that the connecting homomorphism $\partial$ acts by conjugation on $g \in G$. To see this, recall that $\partial$ is generally given on the class of a based loop $\ell \in P_0 X$ by first lifting it through $\mathrm{ev}$ to a based path $\widehat \ell \in P_{[g]} \mathcal{L}X$ and then evaluating that at its endpoint:

$$\partial({[\ell]}) = \big[{\widehat{\ell}_1}\big] \mathrlap{\,.}$$

Here we may take $\widehat{\ell}$ to be given by

$$\widehat{\ell}_t \coloneqq \mathrm{conc}\big({ \overline{\ell}(t-), \gamma, \ell(t-) }\big)$$

(where “$\mathrm{conc}$” denotes concatenation and an overline denotes reversal of paths $[0,1] \to X$), which implies the above equality of exact sequences.

From this, the first claim (1) follows by exactness: The image of $\pi_1(\mathrm{ev})$ is now identified with the kernel of $\mathrm{Ad}_{(-)}(g)$, and that is the centralizer $C_G(g)$, by definition. (Beware here that the copy of “$G$” in the bottom right above is the underlying set of the group, pointed by the element $g$.)

Similarly for the second claim (2): If $\pi_2(X) = \pi_1(\Omega X)$ is trivial, then exactness gives that $\pi_1(\mathrm{ev})$ is injective and hence an isomorphism onto its image.

Example

Given any group $G$, we may consider the Eilenberg-MacLane space $X \coloneqq K(G,1)$. By definition, this has $\pi_1(X) \simeq G$ and $\pi_2(X) \simeq \ast$, and hence Prop. gives that the centralizers of elements of $G$ are obtained as the fundamental groups $\pi_1\big( \mathcal{L}X, \gamma \big) \simeq C_G(g) \mathrlap{\,.}$

Related concepts

• stabilizer subgroup

• normalizer subgroup

References

See also:

• Wikipedia: Centralizer and normalizer