1. ## Groups; solution to quadratic.

I'm new to groups and trying to answer the following question.

Let G be any group. Show that the identity e is the unique solution of the equation $\displaystyle x^2=x$

So how do I proceed?
A group, as far as my primitive understanding goes, is a set governed by a binary operation, producing local inbred offspring; then we have the given axioms: closure, associativity, identity and invertibility.

So we assume G is a group with a binary operation $\displaystyle x*x=x^2$or is it $\displaystyle x^2-1$?

Then what?

we know $\displaystyle x*e=xe-1=x$
and $\displaystyle e*e=e^2-1=e$

Does this help?

What is the best and most comprehensive text on groups that starts at the absolute beginning?

2. You know that elements in a group are always invertible, so what happens when you multiply the identity $\displaystyle x^2=x$ by $\displaystyle x^{-1}$.

As for texts in groups, Rotman's An Introduction to the theory of groups is very good in my opinion.

3. If we let the binary operation be $\displaystyle x*x=x^2$ and we know that $\displaystyle x*e=x$ then (given $\displaystyle x^2=x$) $\displaystyle x*x=x*e$ therefore $\displaystyle x=e$

How do we show it is a unique solution?