Abelianisation of Groups

Subgroup Generated by Subsets

Let be a group and a non-empty subset. The subgroup generated by is

This is indeed a subgroup: it contains (alternatively we could include the empty set in the definition and define the empty product to be 1), it is clearly closed under multiplication and inverses.

e.g. The symmetric group is generated by adjacent transpositions.

Proposition

is the smallest subgroup containing with respect to inclusion.

Commutator Subgroup

Commutator and Commutator Subgroup

The commutator of two elements and in a group G is the element . The commutator subgroup of is the group generated by all commutators:

Proposition

Two elements commute if and only if their commutator is the identity.

Proof This is straightforward from the definition.

Proposition

is a normal subgroup of . Moreover, is abelian.

Proof For any , we have , so is normal. Moreover, for any , we have so is abelian.

Prop

Prop Universal Property of Abelianisation

Centre

Centre

The center of a group is the set of elements that commute with every element of . That is

Proposition

The centre is a normal subgroup of .

Lemma

For any group , is cyclic iff is abelian.

Proof Suppose is abelian, then , so is trivial and hence cyclic. Conversely, suppose is cyclic, then there exists some such that . Let , then there exist integers such that and . Then we have and for some . It follows that Thus is abelian.

Centraliser

Centraliser

The centraliser of an element in a group , is the set of elements that commute with .
Or equivalently, it is the stabilizer of under conjugation.

Derived Series