# proof that $G$ is cyclic if and only if $\delimiter 69640972G\delimiter 86418188=\mathop{exp}\nolimits(G)$

###### Theorem 1

A finite abelian group $G$ is cyclic if and only if $\lvert G\rvert=\exp(G)$.

Proof. $G$ is cyclic if and only if it has an element of order $\lvert G\rvert$. But $\exp(G)$ is the maximum order of any element of $G$. Thus $G$ is cyclic only if these two are equal.

Title proof that $G$ is cyclic if and only if $\delimiter 69640972G\delimiter 86418188=\mathop{exp}\nolimits(G)$ ProofThatGIsCyclicIfAndOnlyIflvertGrvertexpG 2013-03-22 16:34:14 2013-03-22 16:34:14 rm50 (10146) rm50 (10146) 6 rm50 (10146) Proof msc 20A99