Subgroup

The handiest examples live in \((\mathbb{Z},+)\): the multiples \(n\mathbb{Z}\) are subgroups, and in fact they are all of them. In \((\mathbb{R}^*,\cdot)\) the positive reals form a subgroup, and in \(GL_2(\mathbb{R})\) the matrices of determinant \(1\) form another. Every group \(G\) has at least two subgroups, \(\{e\}\) and \(G\) itself, called trivial; the others are called proper.

Checking the full definition would be long, but the subgroup criterion suffices: \(H\subseteq G\) is a subgroup if and only if \(H\ne\varnothing\) and \(ab^{-1}\in H\) for all \(a,b\in H\). If \(H\) is finite, it is enough that it be nonempty and closed under the operation. Note that the identity of \(H\) must be that of \(G\), and closure alone is not enough for infinite sets, as \(\mathbb{N}\subset\mathbb{Z}\) shows.

The intersection of subgroups is always a subgroup, but the union almost never is: \(2\mathbb{Z}\cup3\mathbb{Z}\) contains 2 and 3 but not 5. This gives meaning to the smallest subgroup containing a given set, which is the intersection of all subgroups containing it and is studied next as the generated subgroup.

The size of a subgroup is not arbitrary: Lagrange's theorem will later show that, in a finite group, the order of a subgroup divides the order of the group. So \(\mathbb{Z}_{12}\) has subgroups of orders 1, 2, 3, 4, 6 and 12, one for each divisor, and none of order 5.