Let G be a group with the that there exists a positive integer n such that, for every gG, gn=eG. The exponent of G, denoted expG, is the smallest positive integer m such that, for every gG, gm=eG. Thus, for every finite groupMathworldPlanetmath G, expG divides |G|. Also, for every group G that has an exponent and for every gG, |g| divides expG.

The concept of exponent for finite groups is to that of characterisic (http://planetmath.org/CharacteristicPlanetmathPlanetmath) for finite fieldsMathworldPlanetmath.

If G is a finite abelian group, then there exists gG with |g|=expG. As a result of the fundamental theorem of finite abelian groups (http://planetmath.org/FundamentalTheoremOfFinitelyGeneratedAbelianGroups), there exist a1,,an with ai dividing ai+1 for every integer i between 1 and n such that Ga1an. Since, for every cG, can=eG, then expGan. Since |(0,,0,1)|=an, it follows that expG=an.

Following are some examples of exponents of finite nonabelian groups.

Since |(12)|=2, |(123)|=3, and |S3|=6, it follows that expS3=6.

In Q8={1,-1,i,-i,j,-j,k,-k}, the ring of quaternions of order eight, since |i|=|-i|=|j|=|-j|=|k|=|-k|=4 and 14=(-1)4=1, it follows that expQ8=4.

Since the order of a productPlanetmathPlanetmath of two disjoint transpositionsMathworldPlanetmath is 2, the order of a three cycle is 3, and the only nonidentity elements of A4 are three cycles and products of two disjoint transpositions, it follows that expA4=6.

Since |(123)|=3 and |(1234)|=4, expS412. Since S4 has no elements of order 8, it cannot have an element of order 24. It follows that expS4=12.

Following are some examples of exponents of infinite groups.

Clearly, any infinite group that has an element of infinite order does not have an exponent. On the other hand, just because an infinite group has the that every element has finite order does not that the group has an exponent. As an example, consider G=/, which is a group under addition. Despite that all of its elements have finite order, G does not have an exponent. This is because, for every positive integer n, G has an element of order n, namely 1n+.

On the other hand, some infinite groups have exponents. For example, let 𝔽2 denote the field having two elements. Then 𝔽2[x], the ring of all polynomialsPlanetmathPlanetmath in x with coefficients in 𝔽2, is an abelian groupMathworldPlanetmath under addition. Moreover, it is an infinite group; however, every nonzero element has order 2. Thus, exp𝔽2[x]=2.

Title exponent
Canonical name Exponent
Date of creation 2013-03-22 13:30:21
Last modified on 2013-03-22 13:30:21
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 23
Author Wkbj79 (1863)
Entry type Definition
Classification msc 20A99
Related topic KummerTheory