Generating Set of a Group
Consider a subset \(S\) of a group \(G\), and let \(J\) be the set of subgroups of \(G\) containing S. The group generated by a set \(S\) is defined to be
This is intuitively the smallest (in the sense of inclusions) group which contains \(S\). That is, removing elements from \(\langle S \rangle\) would either mean that it is no longer true that \(S \subseteq \langle S \rangle\) or that \(\langle S \rangle\) is not a group.
This is a group because the intersection of subgroups is a subgroup.
Note that the notation \(\langle a_1, a_2, \dots, a_n \rangle\) is also often used in place of \(\langle \{ a_1, a_2, \dots, a_n \} \rangle\).
The generator for a given group is also not necessarily unique.
\(\langle S \rangle\) can be equivalently defined as the set of elements of the form
where \(s_i \in S\) or \(s_i^{-1} \in S\) for all \(i\) and \(n \geq 0\). When \(n = 0\) we define \(g\) to be the identity.
Intuitively, imagine taking the given set, and adding more elements to it by multiplying the elements together, and taking inverses. Once doing either of these things no longer produces additional elements, the generated group has been constructed.