91Ó°ÊÓ

In each of the following, \(H\) is a subset of \(\mathbb{R} \times \mathbb{R}\) (a) Prove that \(H\) is a normal subgroup of \(\mathbb{R} \times \mathbb{R}\). (Remember that every subgroup of an abelian group is normal.) (b) In geometrical terms, describe the elements of the quotient group \(G / H\). (c) In geometrical terms or otherwise, describe the operation of \(G / H\). $$ H=\\{(x, y): y=-x\\} $$

Let \(G\) be a group, and \(H\) a normal subgroup of \(G\). Prove the following: If \((G: H)=m\), then \(a^{m} \in H\) for every \(a \in G\).