I am taking an Abstract Algebra course and my lecturer is really bad and Im not the only who thinks so. All that we get are his poorly constructed notes and most people in the class are lost. Everyone's thinking of making a complaint.

Anyway, I really want to try and do the exercises he sets us, but I dont have any textbook (we are only given course notes) that I can get examples from and I still need to try and understand the content. I'm hoping that if I can be shown how to answer these 3 questions, I can answer questions that are almost the same but with different number types.

So, can someone please show me how to do the following?

Prove which of the following is (G,*) a semigroup? In which is it a group? If it is a group, is it Abelian?

(1) $\displaystyle G = \mathbb{R} , a$*$\displaystyle b = \frac{a+b}{2} $

(2) $\displaystyle G = \mathbb{N}$ x $\displaystyle \mathbb{R} , (a,x)$*$\displaystyle (b,y) = (a+b, \frac{ax+by}{a+b}) $ and;

(3) $\displaystyle G$ is the set of all strings formed from the letters $\displaystyle a,b, \alpha, \beta $ (including the null string, which contains no letters), with the property that the strings $\displaystyle a \alpha, \alpha a, b \beta, \beta b $ are all equivalent to the null string; so whenever these combinations occur in a string, they may be removed, i.e. $\displaystyle bb \beta \beta abb \alpha a = b \beta abb = abb$.

$\displaystyle A*B$ is the concatenation of $\displaystyle A $ and $\displaystyle B $ i.e. $\displaystyle abb* \beta a = abb \beta a = abb \beta a = aba $

I know its a lot, but I am really stuck with Abstract Algebra.

Now I do know and understand many of the properties of Semigroups, Groups, and Abelian groups but thats because all I get in the lectures are these properties, and rarely ever get examples so that when I get questions like this I dont really know what to do.

I am pretty sure that to do this question, I first have to show whether or not G is empty (what notation can I use for this?), and that I need to show if the operation is well-defined (how do I prove this?). Then I dont know what to do.

. I'm really worried that if I don't learn how to do this now, I'll be stuffed with the remainder of the course's content, and will fail the course

. Can anyone please help??