A group \(G\) is free if it is isomorphic to a free product of copies of \(\Z\text{:}\) i.e., if \(G\cong \prod\limits_{\alpha\in I}^*\Z\) for some set \(I\text{.}\)
Given an indexed set \(\{a_\alpha\}_{\alpha\in I}\text{,}\) we let \(G_\alpha=\langle a_\alpha\rangle\) be the infinite cyclic group with generator \(a_\alpha\) (hence \(G_\alpha\cong \Z\)), and we call \(\prod\limits_{\alpha\in I}^*G_\alpha\cong \prod\limits_{\alpha\in I}^*\Z\) the free group on the elements \(a_\alpha\text{.}\)
Theorem2.14.2.Free groups.
Let \(G\) be a group. The following are equivalent.
\(G\) is a free group.
There is a family \(\{g_\alpha\}_{\alpha\in I}\) of elements of \(G\) such that for all \(g\in G\) we can write
where \(m_k\in \Z\) for all \(1\leq k\leq n\text{;}\) and furthermore this expression is unique if \(\alpha_j\ne \alpha_k\) for all \(j\ne k\) and \(m_k\ne 0\) for all \(1\leq k\leq n\text{.}\)
There is a family \(\{g_\alpha\}_{\alpha\in I}\) of elements of \(G\) such that given any group \(H\) and collection \(\{h_\alpha\}_{\alpha\in I}\) of elements of \(H\text{,}\) there is a unique homomorphism \(\phi\colon G\rightarrow H\) satisfying \(\phi(g_{\alpha})=h_\alpha\text{.}\)
Definition2.14.3.Freely generated.
Let \(G\) be a group. A tuple \((g_\alpha)_{\alpha\in I}\) satisfying the conditions of Theorem 2.14.2 is called a system of free generators. We say that \(G\) is freely generated by the \(g_\alpha\) in this case.
Theorem2.14.4.Quotients of free groups.
Any group \(G\) is isomorphic to a quotient of a free group: i.e., we have \(G\cong F/N\) for some free group \(F\) and normal subgroup \(N\normalin F\text{.}\)
Definition2.14.5.Group presentation.
Let \(F\) be the free group on the elements \(X=\{a_\alpha\}_{\alpha\in I}\text{.}\) Let \(Y\) be an subset of \(F\text{.}\) We denote by \(\langle X\vert Y\rangle\) the quotient \(F/N\text{,}\) where \(N\normalin F\) is the least normal subgroup containing \(Y\text{.}\) We say \(\langle X\vert Y\rangle\) is a presentation of a group \(G'\) with generators \(X\) and relations \(Y\) if \(G'\cong F/N\text{.}\)
Remark2.14.6.
The relations \(Y\) in a group presentation \(\langle X\vert Y\rangle\) should be thought of as elements that get sent to the identity in the quotient \(F/N\text{.}\) For example, if \(abab^{-1}\) is an element of \(Y\) then the element \(\overline{a}\overline{b}\overline{a}\overline{b}^{-1}=e\) in the quotient \(G/N\text{.}\) Equivalently, this means \(\overline{b}\overline{a}=\overline{a}^{-1}\overline{b}\text{.}\) As an abuse of notation, we sometimes include the equality \(ba=a^{-1}b\) among the elements of \(Y\) to emphasize that this relation holds in the quotient. In other words, we write \(\langle a, b\vert ba=a^{-1}b\rangle\) instead of \(\langle a,b\vert abab^{-1}\rangle\)
Example2.14.7.Presentations of groups.
Identify a familiar group with given presentation. Justify your answer.
Let \(G\) be a group. The commutator \([g,h]\) of elements \(g,h\in G\) is defined as \([g,h]=ghg^{-1}h^{-1}\text{.}\) The commutator subgroup of \(G\text{,}\) denoted \([G,G]\) is the subgroup of \(G\) generated by the set of all commutators in \(G\text{.}\)
Theorem2.14.9.Commutator and abelianization.
Let \(G\) be a group.
\([G,G]\) is a normal subgroup of \(G\text{.}\)
The quotient \(G/[G,G]\) is abelian.
Let \(q\colon G\rightarrow G/[G,G]\) be the quotient map. The group \(G/[G,G]\) is the largest abelian quotient of \(G\text{,}\) as defined by the following universal property: if \(A\) is an abelian group and \(\phi\colon G\rightarrow A\) is a homomorphism, there is a unique homomorphism \(\overline{\phi}\colon G/[G,G]\rightarrow A\) satisfying \(\phi=\overline{\phi}\circ q\text{.}\)
Definition2.14.10.Abelianization.
The quotient group \(G/[G,G]\) is called the the abelianization of \(G\text{.}\)
Theorem2.14.11.Abelianization of free group.
Suppose \(G\) is freely generated by the collection \(\{g_\alpha\}_{\alpha\in I}\text{.}\) Let \(q\colon G\rightarrow G/[G,G]\) be the quotient map, and denote \(q(g)=\overline{g}\) for all \(g\in G\text{.}\) The abelianization \(G/[G,G]\) is a free abelian group with basis \(\{\overline{g}_{\alpha}\}_{\alpha\in I}\text{.}\)
Corollary2.14.12.Free groupRank of free group
Let \(G\) and \(H\) be free groups with systems of free generators \(\{g_\alpha\}_{\alpha\in I}\) and \(\{h_\beta\}_{\beta\in J}\text{,}\) respectively. We have \(G\cong H\) if and only if \(\abs{I}=\abs{J}\text{.}\)
Definition2.14.13.Rank of free group.
The rank of a free group \(G\text{,}\) denoted \(\rank G\text{,}\) is the cardinality of any system free generators of \(G\text{.}\)
