# Identity element for group theory

• kala
In summary: All the pieces are there, you just have to put them togetherIn summary, to prove that (S, *) is a group where S is the set of all real numbers except for -1, we must follow the axioms of associativity, identity element, and inverse. The identity element for S is 0, which can be shown by plugging it into the definition for *. To find the inverse, we must solve for an element that satisfies the equations for the identity element. Both of these steps require using ordinary addition and multiplication of real numbers.
kala

## Homework Statement

Show that (S, *) is a group where S is the set of all real numbers except for -1. Define * on S by a*b=a+b+ab

## The Attempt at a Solution

Well I know that i have to follow the axioms to prove this. So I started with G1 which is associativity. This one I got to work. Then G2 says i need to find an identity element. But this is where i got confused. I think that the identity element is 0. but i don't know if that is write. I don't really know how to find it. Then finally G3 is the inverse and I also got stuck here. I don't really know what the inverse would be. Can anyone help.

you just need to work from the definition

so you want to find an element, call it "e", s.t

a*e=e*a. (need to check this holds both ways)

plug that into the definition for *. what i mean is replace the b with e and solve for e. try that. if it works one way you need to show that it works the other way.

Just a note, if this group commutes or is abelian you only need to show one direction, but you would need to show commutativity.

kala said:
I think that the identity element is 0. but i don't know if that is write.
There's an easy way to tell -- plug it into see if it satisfies the identity.

I don't really know how to find it.
Yes you do; you're just being timid. You know the equations that the identity is supposed to satisfy... and you know how to turn it into an ordinary equation involving ordinary addition and ordinary multiplication of real numbers... and you know how to solve equations involving ordinary addition and ordinary multiplication of real numbers... You know all the steps involved, and the path is direct, you just have to start moving.

I don't really know what the inverse would be. Can anyone help.
Again, same situation as the last question.