x*y=x+xyLet the binary operation  be defined on by the given rule. Determine whether is a group with respect to and whether it is an abelian group. State which, if any, conditions fail to hold.

Expert Answers
mlehuzzah eNotes educator| Certified Educator

To be a group, you have a binary operation operating on a set (satisfying some conditions). Here you haven't stated what the set is.

So, I'm going to answer this assuming that the set is `ZZ`, or `QQ`, or `RR`, or `CC`, or something where we have "familiar numbers" and "familiar addition" (no weird definition of "+")

The criteria for a group are:

1) There is an identity element e such that e*x = x*e = x
2) For each x, there is an element, `x^(-1)` such that `x**x^(-1) = x^(-1)**x = e`
3) `(x**y)**z = x**(y**z)`

And for an abelian group:
4) `x**y=y**x`

One at a time:
e*x = e+ex
x*e = x+xe
Assuming by "xe" you mean regular (familiar) multiplication, then ex=xe and these can only be equal if e=x
So this can only happen if there is only 1 element in the group, the identity element. So, unless we have a set with one element, we will fail this condition.

`x**x^(-1) = x + xx^(-1)`
`x^(-1)**x = x^(-1) + x^(-1)x`
This can only happen if, for every x, `x=x^(-1)` and, moreover, `x+xx=e`
Without knowing what set this operator acts on, it is hard to describe what is wrong with this, but this one is likely to fail. For example, if our set contains the natural numbers (for example, any of `ZZ`, `QQ`, etc, but not `ZZ_4` or `ZZ_5`) then we have:
These are not the same, so they can't both be e

So, in the case that our set contains as a subset the natural numbers, this condition fails

`x**(y**z) = x**(y+yz) = x+xy+xyz`
`(x**y)**z = (x+xy)**z = x+xy+xz+xyz`
Unless xz=0, this will be false.
So condition 3 fails

`y**x = y+yx`
Assuming regular multiplication (so xy=yx), then this will only hold true if x=y (for every x and y) which will only hold true in a one-element group. So condition 4 fails

So this is not a group, and it is not an abelian group

mlehuzzah eNotes educator| Certified Educator


The things that look like `xx` should be xx

So, where it says `xx^(-1)`
I meant x times x inverse

And where it says `xx`
I meant x times x