I came across this question, and was wondering what the notation G[n : a] means. Is x the generator? Any comments/explanations will be of great help. The question I found was:
Let $G$ be an abelian group, and let $a\in G$. For $n≥1$, let $G[n:a] := \{x\in G:x^n =a\}$.
(a) Show that $G[n: a]$ is either empty or equal to $αG[n] := \{αg : g \in G[n]\}$, for some $α∈G$. (Recall: $G[n]:=\{x\in G:x^n =1\}$.)
(b) If $G$ is cyclic of order $m$, prove that:
$|G[n:a]| = (n,m)$ if $\text{Ord}(a)\mid(m/(n, m))$, OR $0$ otherwise.