proof that a finite abelian group has element with \delimiter69640972g\delimiter86418188=exp(G)


Theorem 1

If G is a finite abelian group, then G has an element of order exp(G).

Proof. Write exp(G)=piki. Since exp(G) is the least common multiple of the orders of each group element, it follows that for each i, there is an element whose order is a multiple of piki, say |ci|=aipiki. Let di=ciai. Then |di|=piki. The di thus have pairwise relatively prime orders, and thus

|di|=|di|=exp(G)

so that di is the desired element.

Title proof that a finite abelian group has element with \delimiter69640972g\delimiter86418188=exp(G)
Canonical name ProofThatAFiniteAbelianGroupHasElementWithlvertGrvertexpG
Date of creation 2013-03-22 16:34:05
Last modified on 2013-03-22 16:34:05
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 4
Author rm50 (10146)
Entry type Proof
Classification msc 20A99