Centralizer and Normalizer - Properties

Properties

Groups

• The centralizer and normalizer of 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).
• C_G(C_G(S)) contains S, but C_G(S) need not contain S. Containment will occur if st=ts for every s and t in S. Naturally then if H is an abelian subgroup of G, C_G(H) contains H.
• If S is a subsemigroup of G, then N_G(S) contains S.
• If H is a subgroup of G, then the largest subgroup in which H is normal is the subgroup N_G(H).
• 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 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 automorphism group 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 S in Inn(G) is T(N_G(S)), and the subgroup of Inn(G) fixing S is T(C_G(S)).

Rings and algebras

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