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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Contents
Statement
Statement with symbols
Suppose is a group and
is its center. Suppose further that the index
is finite. Let
be a conjugacy class in
; in other words,
is an orbit under the action of the group on itself by conjugation. We then have that
is finite, and further:
.
Related facts
Stronger facts
Weaker facts
Other facts about size of conjugacy class
- Size of conjugacy class is bounded by order of derived subgroup
- Size of conjugacy class need not divide order of derived subgroup
- Size of conjugacy class need not divide index of abelian normal subgroup
Similar facts about degrees of irreducible representations
- Degree of irreducible representation divides index of center
- Degree of irreducible representation divides index of abelian normal subgroup
- Degree of irreducible representation is bounded by index of abelian subgroup
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 , the order of the commutator subgroup is bounded by
.
- Lagrange's theorem: This states that the order of any subgroup divides the order of the group.
- Fundamental theorem of group actions: This states that for any group action, there is a bijection between the orbit of an element and the left coset space of its stabilizer. In particular, it shows that the size of any orbit divides the order of the group.
- Order of quotient group divides order of group: This is a consequence of Lagrange's theorem and the first isomorphism theorem.
- Degree of irreducible representation divides group order
- Degree of irreducible representation divides index of center
- Degree of irreducible representation divides index of abelian normal subgroup
Facts used
- Left coset space of centralizer is in bijective correspondence with conjugacy class
- Index is multiplicative: If
are groups, then
.
Proof
Given: A finite group with center
.
is finite. A conjugacy class
in
.
To prove: is finite and the size of
divides
.
Proof:
Step no. | Assertion/construction | Facts used | Given data/assumptions used | Previous steps used | Explanation |
---|---|---|---|---|---|
1 | Let ![]() ![]() ![]() ![]() |
![]() ![]() | |||
2 | ![]() ![]() ![]() ![]() ![]() |
Fact (2) | Step (1) | Step-fact combination direct. | |
3 | ![]() ![]() |
Fact (1) | Step (2) | By Fact (1), ![]() ![]() |
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