# Groups do not necessarily have to have only one operation

1. Mar 28, 2009

### Gear300

To be sure of things, groups do not necessarily have to have only one operation; they may have more, but there is one and only one operation they are identified with. Am I right?

2. Mar 28, 2009

### CompuChip

Re: Groups

There may be two operations o and * defined on a set G, such that (G, o) and (G, *) are both groups.
When we say "let G be a group" we're sloppy, you should say what the operation is.

3. Mar 28, 2009

### Gear300

Re: Groups

Thus, if I were asked whether the set of all rational numbers Q with addition was a group, would it be valid to assume multiplication as an operation and state that a/b + c/d = (ad + cb)/(bd) ε Q (for a, b, c, d ε Z) in the case of proving that addition is closed with respect to Q.

4. Mar 28, 2009

### CRGreathouse

Re: Groups

Yes, that's valid, but you're not multiplying group elements (which would be bad). You're just expanding the definition of rational addition.

5. Mar 28, 2009

### Gear300

Re: Groups

I see. Thanks for the replies.

6. Mar 28, 2009

### sutupidmath

Re: Groups

Well, in general there is only one operation that is defined in a group. That is there is only one operation which is applied berween the elements of a group.

In your example, we could say let (Q,@) be a group, where Q is the set of rational numbers, and then we would say that @ is defined in this way:

$$\frac{a}{b} @ \frac{c}{d}=\frac{a*d+c*b}{b*d}$$

where + is the natural addition symbol and * multiplication.

But, like it was said above, here you are not multiplying the elements of Q, which we have assumed are of the form

$$Q={ \frac{a}{b}: a,b \in Z }$$

7. Mar 29, 2009

### HallsofIvy

Re: Groups

But in any case, in
$$\frac{a}{b}+ \frac{c}{d}= \frac{ad+ bc}{cd}$$
You are NOT multiplying rational number you are multiplying integers.