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The conjugacy class and centralizer of an element
Definition
Let be a group and (Group and abelian group). The conjugacy class of is
The centralizer of is
The subgroup property implicit in the notation is proved in and are subgroups of ↗.
Depends on
Used by
- A noncentral element of an extraspecial p-group has centraliser of index p Corollary
- Class functions and the complex vector space cf(G) Definition
- The centralizer C_G(H) of a subgroup Definition
- The character table of a finite group Definition
- The class sum Ĉ of a conjugacy class C Definition
- A conjugacy class in an index-two normal subgroup remains one class or splits into two equal classes, with a centralizer criterion Lemma
- A group is abelian exactly when its conjugacy classes are singletons Lemma
- C_G(x) and N_G(H) are subgroups of G Lemma
- Orbit indicators form a basis of invariant functions Lemma
- Every conjugacy class of an extraspecial p-group outside the centre has exactly p elements Proposition
- G/C_G(x)toCl_G(x) is a bijection, so |Cl_G(x)|=[G:C_G(x)] whenever these cardinalities are finite Theorem
- If σ∈ Sₙ has cₖ cycles of length k, then |C_Sₙ(σ)|=∏ₖ₌₁ⁿ k^cₖcₖ! Theorem
- The class equation |G|=|Z(G)|+∑ᵢ [G:C_G(xᵢ)] for a finite group Theorem
Dependency tree · two levels
6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- T. W. Judson, Abstract Algebra: Theory and Applications, 14.2 (standard reference, not scraped)