# Generators of a cyclic group and their orders

a) Let $G = \langle a \rangle$ be a finite cyclic group. Prove that for each $b\in G$, $\langle b \rangle=G$ if and only if order of $b$ equals order of $G$.

b) The previous part does not hold if $G$ is an infinite cyclic group. Why not?

(a) How many elements can $b$ generate? How many elements are in $G$?
• If $<b>$ generates at least $n$ distinct elements then obviously it generates the whole group. If not, then what can you say about $b^{n-1}$ and lower powers of $b$? What does this imply about the order of $b$. – not all wrong May 10 '13 at 11:06