Centralizer and normalizer

Example

Consider the group

${\displaystyle G=S_3=\{[1,2,3],[1,3,2],[2,1,3],[2,3,1],[3,1,2],[3,2,1]\}}$ (the symmetric group of permutations of 3 elements).

Take a subset ${\displaystyle H}$ of the group ${\displaystyle G}$:

${\displaystyle H=\{[1,2,3],[1,3,2]\}.}$

Note that ${\displaystyle [1,2,3]}$ is the identity permutation in ${\displaystyle G}$ and retains the order of each element and ${\displaystyle [1,3,2]}$ is the permutation that fixes the first element and swaps the second and third element.

The normalizer of ${\displaystyle H}$ with respect to the group ${\displaystyle G}$ are all elements of ${\displaystyle G}$ that yield the set ${\displaystyle H}$ (potentially permuted) when the element conjugates ${\displaystyle H}$. Working out the example for each element of ${\displaystyle G}$:

${\displaystyle [1,2,3]}$ when applied to ${\displaystyle H}$: ${\displaystyle \{[1,2,3],[1,3,2]\}=H}$; therefore ${\displaystyle [1,2,3]}$ is in the normalizer ${\displaystyle N_G(H)}$.

${\displaystyle [1,3,2]}$ when applied to ${\displaystyle H}$: ${\displaystyle \{[1,2,3],[1,3,2]\}=H}$; therefore ${\displaystyle [1,3,2]}$ is in the normalizer ${\displaystyle N_G(H)}$.

${\displaystyle [2,1,3]}$ when applied to ${\displaystyle H}$: ${\displaystyle \{[1,2,3],[3,2,1]\}\ne H}$; therefore ${\displaystyle [2,1,3]}$ is not in the normalizer ${\displaystyle N_G(H)}$.

${\displaystyle [2,3,1]}$ when applied to ${\displaystyle H}$: ${\displaystyle \{[1,2,3],[2,1,3]\}\ne H}$; therefore ${\displaystyle [2,3,1]}$ is not in the normalizer ${\displaystyle N_G(H)}$.

${\displaystyle [3,1,2]}$ when applied to ${\displaystyle H}$: ${\displaystyle \{[1,2,3],[3,2,1]\}\ne H}$; therefore ${\displaystyle [3,1,2]}$ is not in the normalizer ${\displaystyle N_G(H)}$.

${\displaystyle [3,2,1]}$ when applied to ${\displaystyle H}$: ${\displaystyle \{[1,2,3],[2,1,3]\}\ne H}$; therefore ${\displaystyle [3,2,1]}$ is not in the normalizer ${\displaystyle N_G(H)}$.

Therefore, the normalizer ${\displaystyle N_G(H)}$ of ${\displaystyle H}$ in ${\displaystyle G}$ is ${\displaystyle \{[1,2,3],[1,3,2]\}}$ since both these group elements preserve the set ${\displaystyle H}$ under conjugation.

The centralizer of the group ${\displaystyle G}$ is the set of elements that leave each element of ${\displaystyle H}$ unchanged by conjugation; that is, the set of elements that commutes with every element in ${\displaystyle H}$. It's clear in this example that the only such element in S₃ is ${\displaystyle H}$ itself ([1, 2, 3], [1, 3, 2]).

Properties

Semigroups

Let ${\displaystyle S^{\prime }}$ denote the centralizer of ${\displaystyle S}$ in the semigroup ${\displaystyle A}$; i.e. ${\displaystyle S^{\prime }=\{x\in A\mid sx=xs\text{ for every }s\in S\}.}$ Then ${\displaystyle S^{\prime }}$ forms a subsemigroup and ${\displaystyle S^{\prime }=S^{‴}=S^{\prime \prime \prime \prime \prime }}$; i.e. a commutant is its own bicommutant.

Groups

Source:

• The centralizer and normalizer of ${\displaystyle S}$ are both subgroups of G.
• Clearly, C_G(S) ⊆ N_G(S). In fact, C_G(S) is always a normal subgroup of N_G(S), being the kernel of the homomorphism N_G(S) → Bij(S) and the group N_G(S)/C_G(S) acts by conjugation as a group of bijections on S. E.g. the Weyl group of a compact Lie group G with a torus T is defined as W(G,T) = N_G(T)/C_G(T), and especially if the torus is maximal (i.e. C_G(T) = T) it is a central tool in the theory of Lie groups.
• C_G(C_G(S)) contains ${\displaystyle S}$, but C_G(S) need not contain ${\displaystyle S}$. Containment occurs exactly when ${\displaystyle S}$ is abelian.
• If H is a subgroup of G, then N_G(H) contains H.
• If H is a subgroup of G, then the largest subgroup of G in which H is normal is the subgroup N_G(H).
• If ${\displaystyle S}$ is a subset of G such that all elements of S commute with each other, then the largest subgroup of G whose center contains ${\displaystyle S}$ is the subgroup C_G(S).
• A subgroup H of a group G is called a self-normalizing subgroup of G if N_G(H) = H.
• The center of G is exactly C_G(G) and G is an abelian group if and only if C_G(G) = Z(G) = G.
• For singleton sets, C_G(a) = N_G(a).
• By symmetry, if ${\displaystyle S}$ and T are two subsets of G, T ⊆ C_G(S) if and only if S ⊆ C_G(T).
• For a subgroup H of group G, the N/C theorem states that the factor group N_G(H)/C_G(H) is isomorphic to a subgroup of Aut(H), the group of automorphisms of H. Since N_G(G) = G and C_G(G) = Z(G), the N/C theorem also implies that G/Z(G) is isomorphic to Inn(G), the subgroup of Aut(G) consisting of all inner automorphisms of G.
• If we define a group homomorphism T : G → Inn(G) by T(x)(g) = T_x(g) = xgx⁻¹, then we can describe N_G(S) and C_G(S) in terms of the group action of Inn(G) on G: the stabilizer of ${\displaystyle S}$ in Inn(G) is T(N_G(S)), and the subgroup of Inn(G) fixing ${\displaystyle S}$ pointwise is T(C_G(S)).
• A subgroup H of a group G is said to be C-closed or self-bicommutant if H = C_G(S) for some subset S ⊆ G. If so, then in fact, H = C_G(C_G(H)).

Rings and algebras over a field

Source:

• Centralizers in rings and in algebras over a field are subrings and subalgebras over a field, respectively; centralizers in Lie rings and in Lie algebras are Lie subrings and Lie subalgebras, respectively.
• The normalizer of ${\displaystyle S}$ in a Lie ring contains the centralizer of ${\displaystyle S}$.
• C_R(C_R(S)) contains ${\displaystyle S}$ but is not necessarily equal. The double centralizer theorem deals with situations where equality occurs.
• If ${\displaystyle S}$ is an additive subgroup of a Lie ring A, then N_A(S) is the largest Lie subring of A in which ${\displaystyle S}$ is a Lie ideal.
• If ${\displaystyle S}$ is a Lie subring of a Lie ring A, then S ⊆ N_A(S).