Subgroups Generated by Subsets

Subgroup Generated by Subsets

Let $G$ be a group and $X\subset G$ a non-empty subset. The subgroup generated by $X$ is $⟨X⟩:=\{g\in G\mid \exists n\in {\mathbb{N}}_0,x_i\in X,e_i=\pm 1,\mathrm{such}\mathrm{that}g=x_1^{e_1}\cdots x_n^{e_n}\}$ This is indeed a subgroup: it contains $1=x^{-1}x$ (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.

Prop It is the smallest subgroup containing $X$ with respect to inclusion.

Commutator and Commutator Subgroup

The commutator of two elements $a$ and $b$ in a group G is the element $[a,b]=ab{a}^{-1}{b}^{-1}$. The commutator subgroup of $G$ is the group generated by all commutators: $[G,G]:=⟨[a,b]:a,b\in G⟩$

Prop If two elements do not commute, their commutator is not the identity.

Prop $[G,G]$ is a normal subgroup of $G$.

Prop $G_{\text{ab}}:=G/[G,G]$ is abelian.

Prop Universal Property of Abelianisation

Centre

Centre

The center of a group $G$ is the set of elements that commute with every element of $G$. That is $Z(G)=C_G=\{z\in G\mid \forall g\in G,zg=gz\}$

Prop The centre is a normal subgroup of $G$: $C_G◃G$

Lemma For any group $G$, $G/Z(G)$ is cyclic iff $G$ is abelian.