Size of conjugacy class divides index of center

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This article gives the statement, and possibly proof, of a constraint on numerical invariants that can be associated with a finite group

This article states a result of the form that one natural number divides another. Specifically, the (size) of a/an/the (conjugacy class) divides the (center) of a/an/the (index of a subgroup).
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Statement with symbols

Suppose G is a group and Z(G) is its center. Suppose further that the index [G:Z(G)] is finite. Let K be a conjugacy class in G; in other words, K is an orbit under the action of the group on itself by conjugation. We then have that K is finite, and further:

|K| | [G:Z(G)].

Related facts

Stronger facts

Weaker facts

Other facts about size of conjugacy class

Similar facts about degrees of irreducible representations

Facts about groups in which the index of the center is finite

A group where the center has finite index is termed a FZ-group, while a group where the conjugacy class of every element is finite in size is termed a FC-group. This result implies that every FZ-group is a FC-group.

FZ-groups have a number of interesting properties. For instance, the Schur-Baer theorem asserts that the derived subgroup of a FZ-group is finite. In fact, if the center has index n, the order of the commutator subgroup is bounded by n^{2n^3}.

Other related facts

Facts used

  1. Left coset space of centralizer is in bijective correspondence with conjugacy class
  2. Index is multiplicative: If A \le B \le C are groups, then [C:A] = [C:B][B:A].


Given: A finite group G with center Z(G). [G:Z(G)] is finite. A conjugacy class K in G.

To prove: K is finite and the size of K divides [G:Z(G)].


Step no. Assertion/construction Facts used Given data/assumptions used Previous steps used Explanation
1 Let g \in K and L = C_G(g). Then, L contains Z(G) Z(G) is the set of elements that commute with every element, therefore, it is contained in C_G(g) = L.
2 L is a subgroup of finite index in G and [G:Z(G)] = [G:L][L:Z(G)], so in particular the index of L divides [G:Z(G)] Fact (2) Step (1) Step-fact combination direct.
3 |K| is finite and divides [G:Z(G)] Fact (1) Step (2) By Fact (1), |K| = [G:L]. By Step (2), the latter divides [G:Z(G)], hence so does the former.
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