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 or derived subgroup of is the group generated by all commutators:

e.g. For the symmetric group , we have .

Remark

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

Proposition

is the smallest normal subgroup of , so that is abelian. This means, is abelian if and only if is trivial.

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

Prop Universal Property of Abelianisation

Derived Series

The derived series of a group is a series of normal groups by repeatedly taking commutator subgroup:

Thus .
A group is solvable if its derived series reaches the trivial group after finitely many steps.

e.g.

• The smallest non-solvable group is the alternating group .
• Any abelian group is solvable.
  

Proposition

A group is solvable iff there exists a finite sequence of normal subgroups:

such that each is normal in , and each quotient ​ is abelian.

Proposition

If is solvable, any quotient of is solvable.

Proposition

A simple solvable group must be cyclic of a prime order.

Proof Suppose is both simple and solvable. Then can be either trivial or the whole group. But can not be because is solvable, so it is trivial. This means is abelian. Finally, note that any finite simple abelian group is cyclic with a prime order.

Corollary

Every nontrivial finite solvable group has a normal subgroup of prime index.

Proof Suppose is finite and solvable. We choose a maximal normal subgroup . Then is simple and solvable, so is cyclic with a prime order . is then the desired normal subgroup.

Feit–Thompson Theorem

Every finite group of odd order is solvable.

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.

Lower Central Series and Nilpotency

The lower central series of a group is defined recursively:

A group is nilpotent if for some .