Group and semigroup The
centralizer of a subset
S of group (or semigroup)
G is defined as :\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\}, 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 S=\{a\} is a
singleton set, we write C
G(
a) instead of C
G({
a}). Another less common notation for the centralizer is Z(
a), which parallels the notation for the
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 :\mathrm{N}_G(S) = \left\{ g \in G \mid gS = Sg \right\} = \left\{g \in G \mid gSg^{-1} = S\right\}, where again only the first definition applies to semigroups. If the set S is a subgroup of G, then the normalizer N_G(S) is the largest subgroup G' \subseteq G where S is a
normal subgroup of G'. The definitions of
centralizer and
normalizer are similar but not identical. If
g is in the centralizer of
S and
s is in
S, then it must be that , but if
g is in the normalizer, then for some
t in
S, with
t possibly different from
s. That is, elements of the centralizer of
S must commute pointwise with
S, 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
normal closure. Clearly C_G(S) \subseteq N_G(S) and both are subgroups of G.
Ring, algebra over a field, Lie ring, and Lie algebra If
R is a ring or an
algebra over a field, and
S is a subset of
R, then the centralizer of
S is exactly as defined for groups, with
R in the place of
G. If \mathfrak{L} is a
Lie algebra (or
Lie ring) with Lie product [
x,
y], then the centralizer of a subset
S of \mathfrak{L} is defined to be :\mathrm{C}_{\mathfrak{L}}(S) = \{ x \in \mathfrak{L} \mid [x, s] = 0 \text{ for all } s \in S \}. The definition of centralizers for Lie rings is linked to the definition for rings in the following way. If
R is an associative ring, then
R can be given the
bracket product . Of course then if and only if . If we denote the set
R with the bracket product as L
R, then clearly the
ring centralizer of
S in
R is equal to the
Lie ring centralizer of
S in L
R. The Lie bracket can also be viewed as an operation of the set
\mathfrak{L} on itself, because
[*,*]:\mathfrak{L}\times\mathfrak{L}\rightarrow\mathfrak{L}. The Lie bracket makes
(\mathfrak{L},[*,*]) a group and its centralizer would then be all elements \{ x \in \mathfrak{L} \mid [x, s] = [s, x] \text{ for all } s \in S \}. However, since the Lie bracket is alternating, this condition is equivalent to \{ x \in \mathfrak{L} \mid [x, s] = 0 \text{ for all } s \in S \}. Thus, the centralizer is defined in the same way for Lie algebras as for groups. The normalizer of a subset
S of a Lie algebra (or Lie ring) \mathfrak{L} is given by :\mathrm{N}_\mathfrak{L}(S) = \{ x \in \mathfrak{L} \mid [x, s] \in S \text{ for all } s \in S \}. While this is the standard usage of the term "normalizer" in Lie algebra, this construction is actually the
idealizer of the set
S in \mathfrak{L}. If
S is an additive subgroup of \mathfrak{L}, then \mathrm{N}_{\mathfrak{L}}(S) is the largest Lie subring (or Lie subalgebra, as the case may be) in which
S is a Lie
ideal. ==Example==