I can see that it is true, but not sure how to prove it...

Problem:

Let G be a group of order 3, and let, e,a,b, be the three elements of G, where e is the identity of the group. SHOW that b must be the inverse of a.

TIA

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- Oct 15th 2007, 05:48 PMginafaraNeed some help with groups..
I can see that it is true, but not sure how to prove it...

Problem:

Let G be a group of order 3, and let, e,a,b, be the three elements of G, where e is the identity of the group. SHOW that b must be the inverse of a.

TIA - Oct 16th 2007, 05:45 AMtopsquark
G is closed and each of its elements are distinct so $\displaystyle ab$ is equal to e, a, or b. Well $\displaystyle ab = a \implies b = e$ and $\displaystyle ab = b \implies a = e$, so we are left with $\displaystyle ab = e \implies a^{-1} = b$. (Similarly for $\displaystyle ba$.)

-Dan