Editing Centralizer and normalizer (section)

===Group and semigroup===
The '''centralizer''' of a subset ''<math>S</math>'' of group (or semigroup) ''G'' is defined as<ref>Jacobson (2009), p. 41</ref>

:<math>\mathrm{C}_G(S) = \left\{g \in G \mid gs = sg \text{ for all } s \in S\right\} = \left\{g \in G \mid gsg^{-1} = s \text{ for all } s \in S\right\},</math>

where only the first definition applies to semigroups.
If there is no ambiguity about the group in question, the ''G'' can be suppressed from the notation. When <math>S=\{a\}</math> is a [[singleton (mathematics)|singleton]] set, we write C<sub>''G''</sub>(''a'') instead of C<sub>''G''</sub>({''a''}). Another less common notation for the centralizer is Z(''a''), which parallels the notation for the [[Center (group theory)|center]].  With this latter notation, one must be careful to avoid confusion between the '''center''' of a group ''G'', Z(''G''), and the ''centralizer'' of an ''element'' ''g'' in ''G'', Z(''g'').

The '''normalizer''' of ''S'' in the group (or semigroup) ''G'' is defined as

:<math>\mathrm{N}_G(S) = \left\{ g \in G \mid gS = Sg \right\} = \left\{g \in G \mid gSg^{-1} = S\right\},</math>

where again only the first definition applies to semigroups. If the set <math>S</math> is a subgroup of <math>G</math>, then the normalizer <math>N_G(S)</math> is the largest subgroup <math>G' \subseteq G</math> where <math>S</math> is a [[normal subgroup]] of <math>G'</math>. The definitions of ''centralizer'' and ''normalizer'' are similar but not identical. If ''g'' is in the centralizer of ''<math>S</math>'' and ''s'' is in ''<math>S</math>'', then it must be that {{nowrap|1=''gs'' = ''sg''}}, but if ''g'' is in the normalizer, then {{nowrap|1=''gs'' = ''tg''}} for some ''t'' in ''<math>S</math>'', with ''t'' possibly different from ''s''. That is, elements of the centralizer of ''<math>S</math>'' must commute pointwise with ''<math>S</math>'', but elements of the normalizer of ''S'' need only commute with ''S as a set''. The same notational conventions mentioned above for centralizers also apply to normalizers. The normalizer should not be confused with the [[conjugate closure|normal closure]].

Clearly <math>C_G(S) \subseteq N_G(S)</math> and both are subgroups of <math>G</math>.