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## Homework Statement

Let G be a nonempty

**finite**set with an associative binary operation such that:

for all a,b,c in G

ab = ac => b = c

ba = ca => b = c

(left and right cancellation)

Prove that G is a group.

**2. The attempt at a solution**

Let a [itex]\in[/itex] G, the set <a> = {[itex]a^k[/itex] : k [itex]\in[/itex] N} is a finite closed subset of G. So, [itex]\exists[/itex]([itex]k_1[/itex], [itex]k_2[/itex])[itex]\in[/itex]N, such that:

[itex]a^{k_1}=a^{k_2}[/itex] using the cancellation property I found that [itex]a=a.a^{k_2-k_1}[/itex].

So,

[itex]a^{k_2-k_1}[/itex] is the identity but the problem is in this reasoning every cyclic subgroup will generate a different identity. And the identity is supposed to be unique. I don't know how to proceed now, any help would be appreciate it.

Thanks is advance,

NAta