What is the exponent of a group?

Solution 1:

The definition of the exponent is not completely correct. The exponent of a group $G$ is the non-negative generator of the ideal $\{z \in \mathbb{Z} : \forall g \in G (g^z=1)\}$. That means: Either it is zero (usually people then say that the exponent is infinite ...), or it is positive, and then it is the smallest positive natural number $z$ such that $g^z=1$ for all $g \in G$.

What's the point? First of all, the exponent is an isomorphism invariant of a group, meaning that two isomorphic groups have the same exponent. This means that the class of groups (up to isomorphism) decomposes into classes of groups (up to isomorphism) of given exponent $e$, for every $e \geq 0$. This can be useful for the classification of groups.

Easy examples: Groups of exponent $1$ are trivial. Groups of exponent $2$ are abelian (this is a standard exercise). Groups of exponent $3$ are not necessarily abelian, as the Heisenberg group over $\mathbb{F}_3$ shows.

Burnside's problem is to find those positive natural numbers $n,m$ such that every $m$-generated group of exponent $n$ is finite. It is an open problem. See here for the solved case $n=3$.