Give an example where H is not a subgroup.

[hsong9]

Homework Statement

If G is an abelian group, show that H = { a in G | a^2=1} is a subgroup of G.
Give an example where H is not a subgroup.

The Attempt at a Solution

For showing H is a subgroup of G, hh' in G and h^-1 in G.
(a^2)(a^2) in G also a = a^-1 in G so H is a subgroup of G.. right?

counterexample is Z+_(4) = {0,1,2,3,4}.. right?

Thanks

 

[MathematicalPhysicist]

To show that H is a subgroup you need to show that it's closed under multiplication under H, take h,h' in H then you need to show that (hh')^2=1 which is game in the pond.
The same goes for h^-1 and 1.

For an example where Z+_(4) well if it's Z/4Z then no, cause it doesn't satisfy a+a=0.

 

[hsong9]

so.. If a = hh', then hh' in H
and (hh')^2 = a^2 = 1 in H... ?

 

[MathematicalPhysicist]

Not really, (hh')^2=hh'hh'=h^2h'^2 for the last equality I used G being abelian.