# Size of conjugacy class divides index of center

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

### 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

### 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 $n^{2n^3}$.

## Facts used

1. Size of conjugacy class equals index of centralizer
2. Index is multiplicative: If $A \le B \le C$ are groups, then $[C:A] = [C:B][B:A]$.

## Proof

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 $[G:Z(G)]$.

Proof: Follows mostly directly from facts (1) and (2), and the observation that the center is contained in the centralizer of any element. PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]