Finite group of prime exponent has prime-power order
= Finite group of prime exponent has prime-power order
{title2=$\exp G=p\ \Longrightarrow\ |G|=p^n$}
If every nonidentity element of a <finite group> has order equal to a prime $p$, its order is a power of $p$. For any prime divisor $q$ of the <group> order, <Cauchy theorem for groups> gives an element of order $q$, forcing $q=p$. This statement does not imply that the <group> is abelian; odd-characteristic <upper unitriangular groups> provide counterexamples.