# Commutator subgroup and center

• ehrenfest
In summary, the center of a group does not always contain the commutator subgroup, as this would imply that commutators commute which is not always the case. However, there is a theorem that states if a normal subgroup N contains the commutator subgroup C, then the quotient group G/N is abelian. This does not guarantee that G/Z(G) is abelian, as seen in the example of Z(A_5) being trivial and A_5/Z(A_5) being non-abelian.
ehrenfest

## Homework Statement

Please confirm that the center of a group always contains the commutator subgroup. I am pretty sure its true.

## The Attempt at a Solution

It's not. That would imply commutators commute, which there's no good reason to expect, and it should be easy to find a counterexample.

But my book has a theorem that says:

"If N is a normal subgroup of G, then G/N is abelian if and only if C is contained in N"
where C is the commutator subgroup.

Clearly the center is normal and its quotient group is abelian!

G/Z(G) is not necessarily abelian, e.g. Z(A_5) is trivial (since A_5 is simple), whence A_5/Z(A_5) =~ A_5, a non-abelian group.

## What is the commutator subgroup of a group?

The commutator subgroup of a group G is the subgroup generated by all the commutators [a,b] where a and b are elements of G. It is denoted by G'.

## What is the center of a group?

The center of a group G is the set of elements that commute with every other element in G. It is denoted by Z(G) or C(G).

## How are the commutator subgroup and center related?

The commutator subgroup is a subgroup of the center. It is the smallest normal subgroup of G such that the quotient group G/G' is abelian. In other words, the commutator subgroup measures how "non-abelian" a group is, while the center measures how "central" a group is.

## What is the significance of the commutator subgroup and center in group theory?

The commutator subgroup and center are important concepts in group theory as they provide information about the structure and properties of a group. They are used to classify groups into different types, such as abelian and non-abelian groups. They also play a crucial role in the study of group homomorphisms and group extensions.

## Can the commutator subgroup and center be equal?

Yes, the commutator subgroup and center can be equal for certain types of groups. For example, all abelian groups have trivial commutator subgroup and center, while some non-abelian groups such as the quaternion group have non-trivial commutator subgroups and centers that are equal.

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