Is 7xy a group in the set R?

Question

Hey,

So i think i'm getting to grips with this but just need to check my logic is correct here.

Is the following a group, The set of all real number R, with the binary operation 7xy.

It appears to meet the assositivity requirment as (x y)z = 49xyz = x(y z) and this is in the set of real numbers

The identity element is 1/7, which is also fine as also in set of real numbers

However it fails on the inverse test, as we end up with y=1/(49x) so if x=0 this can not hold.

So 7xy in the set of R, is not a group.

However it would be a group in the set R{0}, as zero is excluded?

Does this logic appear correct.

Thank you Avery ;-)

Answer 1

What you say is correct. The zero element 0 can not have an inverse and that is why you have to restrict the set to R-{0}.