Modern Abstract

Modern Abstract Aljebra: proof?

Let G be a group with a finite number of elements. Show that for any [element] a in G there exists an n in Z*(positive integer) such that a^n = e.

author's hint: consider e,a,a^2,a^3... a^m for m elements in G and then use the cancellation laws.

Fix an element a in G. Consider the set S={a,a^2,a^3,...}. By the closure property, S is a subset of G. Any subset of a finite set is finite, so S is finite. Therefore the elements a, a^2, a^3, ... cannot all be distinct. Hence there must be some a^j and a^k such that j

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