Algebra Theory?

Question

1. In each case, determine whether * is associative, whether * has an identity element and, if the latter is the case, whether each element of S has an inverse (with regards to *), ie whether S is a group:
(a) S={1,-1}, * ordinary multiplication of integers
(b) S= the subset of Q consisting of all positive rationals that have a rational square roots, *=ordinary multiplication of rationals
(c) S=R\{0}, the set of all non-zero reals with a*b=|ab|

2. From examples above that are groups, which are abelian?

3. Consider groups Z(subscript n) with binary operation being addition modulo n. Write down operation tables for
(a) n=4,5
(b) n=6

4.Prove that if G=, then G=

5. Let G be a finite group, and let g,x∈ G
(a) prove that x and g^-1xg have the same order
(b) Let a,b∈G. Prove that ab and ba have the same order

This is exam revision. I really appreciate the help. if possible i need full working out and explinations.

Thanks

Don

Answer 1

Here's a start:

On 1a and 1b -- if something is associative, and you only look at a subset of where it's defined, that doesn't stop it from being associative. (Ditto commutative.)

As to whether there's an identity -- well, is 1 in the set or not? The inverses question should also be pretty easy.

On 1c -- is there any k such that -1 * k = -1? I think not. So there's no identity.

2 -- I answered above. :)

3 -- if you can't do this yourself, you need more help than we can give you here.

4 and 5 are more interesting. Why don't you go ahead and reask them as separate questions?