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I don't really understand the definition:

The exponent of a group G is the smallest natural number x such that for all $g \in G,g^x = e$.

It seems like it's saying, for EVERY element of the group, when you keep applying the group operation to itself which power to itself gives you e.What is the lowest number that this is true for for all elements of G.

First of all, what would even be the point of creating some definition like that, what purpose does something like this serve? I guess, I would see that you could get the lcm of all the exponents that equal e, but it seems like a pretty tedious process to figure out where g's equal e.

I am obviously missing something, can someone help me out here?

Thanks scores.

pink floyd
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George Stephnopolis
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    The point is that the definition can be used to prove things (or rather, to express provable things more succinctly). For example. Every group of exponent $2$ is abelian. - The exponent of a finite group divides its order. - The order of a finite group divides a suitable power of its exponent – Hagen von Eitzen Apr 10 '15 at 12:52
  • It's particularly useful to prove that every finite subgroup of the group of units of a field is cyclic. Maybe I didn't get your precise question. – Daniel Apr 10 '15 at 12:52
  • First, there are groups with infinite exponent even that they're torsion group's. Second, read a little about this in group theory books to see what they use it for,. – Timbuc Apr 10 '15 at 12:52
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    Note that the definition of exponet through the l.c.m. works with abelian groups, but generally not with non-abelian ones. – Bernard Apr 10 '15 at 13:20

1 Answers1

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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 $n=3$.

Martin Brandenburg
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