
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
b) {1,0,1}, addition
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???

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: (23)4=5 and 2(34)=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.