Understanding Inverses & Max* for Integers

[kathrynag]

Homework Statement

If I have a group defined on the integers, by a*b=ab, how do I know if an inverse exists?
Also, define * on the integers by a*b=max{a,b}

Homework Equations

The Attempt at a Solution

I got 1/a as an inverse, but I'm thinking it's not a group since we don't know if 1/a is an element of the integers.

The max is confusing me.
I know I need to check associativity, identity, and inverse for a group.
a*(b*c)=a*max{b,c)
=max{a,max{b,c}}
(a*b)*c=max{a,b}*c
=max{max{a,b},c}
I'm already confused at this point.

 

[lanedance]

you only need contradict one of the properties for it not to be a group

The first one is correct, for a not equal to e, the inverse does not exist within the group.

For the 2nd
associativity is ok, as max{max{a,b},c} = max{a,b,c}
you could try the identity, whici si clearly not unique
or probably more straight forward show there is no inverse, consider a*b with a<b