1. ## forms of groups

how do I tell wether the given se of numbers forms a group with respect to the given operation?

a) {-1,1}, multiplication
c) Z, subtraction
d) {2^m : m is a member of Z}, multiplication
e) the set of matrices ,... a,b is a member of R, (a^2) + (b^2) >0, multiplication

a b
-b a

I think you have to tell if they are associative or not, and I think (a) is, but how do you know for sure???

2. For the finite sets, all you have to do is check each element. In {-1,1}, 1 is the identity, -1 is it's own inverse, and by checking each possible product of those two elements, it's easy to see that the set has closure. Also, since -1 and 1 are real numbers, and multiplication is associative for all reals, the associative property obviously holds for this set.

The second set fails the closure axiom, because 1+1=2 and 2 is not an element of the set.

The third set fails associativity. This is easy to see with an example: (2-3)-4=-5 and 2-(3-4)=3.

You should be able to handle the other two. If you suspect something is not a group, look for a counterexample. Otherwise, try and verify each group axiom.