Let $(G, ∗)$ be a group, let $H$ be a normal subgroup of $G$, and let $(G/H, ⋆)$ denote the quotient group of $G$ by $H$.
(a) Prove that if $xH \in G/H$, then $(xH)^m = x^mH$ for all $m ∈ Z$.
(b) Suppose that there is no positive integer $m$ for which $x^m ∈ H$. Prove that $ord(xH) = ∞$.
(c) Suppose that there exists a positive integer $m$ for which $x^m ∈ H$. Prove that $ord(xH) = d$, where $d$ is the least positive integer for which $x^d ∈ H$.