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centralizer and normalizer

 Definition/Summary The centralizer C(a) of an element a of G is the set of all elements of G that commute with a. The centralizer C(H) of a subgroup H of G is the set of all elements of G that commute with every element of H. The normalizer N(H) of a subgroup H of G is the set of all elements of G under which H is self-conjugate. The center of a group is the group's centralizer, while a group's normalizer is itself.

 Equations Centralizer of element a: $C(a) = \{g \in G | ga = ag \}$ The order of an element's centralizer and the order of its conjugacy class multiply to get the order of the group. Centralizer of subgroup H: $C(H) = \{g \in G | gh = hg ,\ \forall h \in H \}$ Normalizer of subgroup H: $N(H) = \{g \in G | gh = h'g ,\ h' \in H ,\ \forall h \in H \}$

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 Breakdown Mathematics > Algebra >> Group Theory