Exponent of a finite group

The exponent of a finite group is the least common multiple of the orders of all its elements.

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.

Every finite subgroup of $K^{\times }$ is cyclic. If its exponent is $m$, all its elements are roots of $X^m-1$, so its size is at most $m$; finite abelian group theory supplies an element of order $m$, forcing equality.