• Sep 11th 2010, 09:50 AM
berachia
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?
• Sep 11th 2010, 09:02 PM
Jose27
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.
• Sep 12th 2010, 05:17 AM
berachia
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?