Help clarifying a question regarding (i think) cyclic groups

[bennyska]

Homework Statement

Let G be a group with a finite number of elements. Show that for any a in G, there exists an n in Z+ such that aⁿ=e.

Homework Equations

a hint is given: consider e, a, a²,...a^m, where m is the number of elements in G, and use the cancellation laws.

The Attempt at a Solution

so part of the trouble I'm having (i'm guessing the most important part), is figuring out what they're asking. is the number n I'm looking for a number that when any a in G is raised to, it gives the identity? i.e. (i'm just picking random letters) aⁿ= e and bⁿ=e when a and b are not equal but n is the same in both? i was working on this question for a while thinking it to mean that a particular a^p might have n₀ while a^q might have n₁ to take it to e (identity).

 

[bennyska]

and then another problem in the same set as the previous one, that may hinge on the same language:
show that if G is a finite group with identity e and with an even number of elements, then there is a not equal to e such that a*a=e.
does this mean every element in G is its own inverse, or just one particular element? not really sure where i'd begin either way, but just to point me in the right direction.