centralizer

Fix an element of a group and ask: which other elements commute with it — that is, don't care in which order you multiply? The set of all such cooperative elements is its centralizer. It measures how “central” the chosen element is: the bigger the centralizer, the more of the group treats that element as interchangeable in products.

Formally, the centralizer in G of an element a is C_G(a) = { g in G : ga = ag }, and more generally the centralizer of a subset S is C_G(S) = { g in G : gs = sg for all s in S }. Each centralizer is a subgroup of G. The centralizer of the whole group is the center Z(G), the elements commuting with everything.

Centralizers control conjugacy: the centralizer C_G(a) is precisely the stabilizer of a under the conjugation action, so by orbit-stabilizer the conjugacy class of a has size [G : C_G(a)]. Thus large centralizer means small class and vice versa. Centralizers also appear constantly in the structure theory of finite groups — for instance, in a finite simple group the centralizer of an involution carries enormous structural information, a fact at the heart of the classification.