Determining a group, by checking the group axioms

[umzung]

Homework Statement

1. For the following sets, with the given binary operation, determine whether or not it forms a group, by checking the group axioms.

Homework Equations

1. (R,◦), where x◦y=2xy+1
   
   
2. (R*,◦), where x◦y=πxy and R* = R - {0}
   

The Attempt at a Solution

For question 1, I found a G2 identity to be 1/2 - 1/(2x), meaning x cannot be zero and therefore (R,◦) is therefore not a group. Is this in the right ballpark?

For question 2, I found a G2 identity to be 1/π, and a G3 inverse to be 1/(π²x), (since x does not equal zero). G1 is closed and G4 is associative, so (R*,◦) is a group.[/B]

 

[andrewkirk]

For question 1, I found a G2 identity to be 1/2 - 1/(2x), meaning x cannot be zero and therefore (R,◦) is therefore not a group. Is this in the right ballpark?

It's not so much that x cannot be zero as that the identity varies between elements - i.e. it depends on x. To satisfy the identity axiom (what you are calling G2), the identity must be the same for all elements. That is, it must be a single element e such that, for all $x$, $e\circ x=x\circ e = x$.