Math 2.0

Group theory and generalizations Definition

Normaliser

Builds on SubgroupConjugate subgroup

Definition

Let GG be a group and let SS be a subset of GG. The normaliser of SS in GG is the set NG(S)={g∈G:gSg−1=S}N_G(S) = \{g \in G : gSg^{-1} = S\}, where gSg−1={gsg−1:s∈S}gSg^{-1} = \{gsg^{-1} : s \in S\}.

Remarks

The normaliser NG(S)N_G(S) is a subgroup of GG, and it contains the centraliser CG(S)C_G(S). If HH is a subgroup of GG, then HH is a normal subgroup of NG(H)N_G(H). Also, NG(H)N_G(H) is the largest subgroup of GG with this property. Thus HH is normal in GG exactly when NG(H)=GN_G(H) = G. The number of conjugates of HH in GG is the index [G:NG(H)][G : N_G(H)].

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Group, concept, no evidence levelGroupconceptConjugate elements, definition, no evidence levelConjugate elementsdefinitionSubgroup, definition, no evidence levelSubgroupdefinitionConjugate subgroup, definition, no evidence levelConjugate subgroupdefinitionNormaliser, definition, no evidence levelNormaliserdefinition
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  1. Group concept
  2. Conjugate elements definition
  3. Subgroup definition
  4. Conjugate subgroup definition
  5. Normaliser definition

Where this leads

9 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Centralizer_and_normalizer, Group and semigroup
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DefinesNormaliser
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