Question

Let * be defined in Z by m*n=m+n+2 Show that (Z,*) is an abelian group?

Answer 1

For`(Z,"*")` to be an Abelian group the following need to hold true-for a group:
1) Closure
2) Associative property
3) Identity element
4) inverse elements

in addition for an Abelian group

5) commutative property

Explanation:

The 5 axioms will be dealt with separately.

`(Z,"*")" where "m,ninZZ`

`m"*"n=m+n+2`

1) closure

ie.`" "AAm,ninZZ, m "*"ninZZ`

`because (m+ninZZ) nn(2inZZ)=>m+n+2inZZ`

closure holds

2) Associatve property

ie `" " m " * "( n " * "p) = (m " * " n) " * "p`

take LHS

`" "m"* " (n " * "p)=m "* "(n+p+2)`

`=m+n+p+2=(m+n+2) +p`

`=(m " * "n)" * "p`

associative property holds

3) identity element

ie `" "EEe :" " m" * "e=e" * "m=m`

`m" * "e=m +e+2=m`

`=>e=-2`

so identity exists. In fact for groups the identity can be shown to be unique.

the second half of the identity

`e " * "m`

to demonstrate is left for the reader to try.

4) Inverses

ie `" " AAinZZ" " EEm^(-1):" "m " * "m^(-1)= m^(-1) "* "m=e`

once more only the first part will be done, and the reader can do the second.

`m " * "m^(-1)=-2`

`m+m^(-1)+2=-2`

`=>m^(-1)=-4-m`

so inverses exist

This just leaves the Abelian property

5) Commutative

ie`" " AAm,ninZZ" "m "* "n=n " * "m`

`m " * "n=m+n+2`

`because m,ninZZ, m+n=n+m`

`:. m+n+2=n+m+2=n " * "m`

so the commutative property holds

we conclude that `(Z ,"* ")` is an Abelian Group