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centralizer and normalizer
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Definition/Summary
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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. |
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Equations
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Centralizer of element a:
[itex]C(a) = \{g \in G | ga = ag \}[/itex]
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:
[itex]C(H) = \{g \in G | gh = hg ,\ \forall h \in H \}[/itex]
Normalizer of subgroup H:
[itex]N(H) = \{g \in G | gh = h'g ,\ h' \in H ,\ \forall h \in H \}[/itex] |
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Recent forum threads on centralizer and normalizer
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Breakdown
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Mathematics
> Algebra
>> Group Theory
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Commentary
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