Show HK are subgroups of a direct product G, HK=KH=G

[kathrynag]

Homework Statement

Let G1 and G2 be groups and let G be the direct product G1 x G2.
Let H={(x1,x2) in G1 x G2 such that x2=e} and let K={(x1,x2) in G1 x G2

such that x1=e}
a) Show H and K are subgroups of G
b) Show HK=KH=G
c) Show that H intersect K={(e,e)}

Homework Equations

The Attempt at a Solution

a) We can further define H and K as (x1,e) and (e,x2)
We want (x1,e) and (e,x2) to be subgroups of (x1,x2). I understand
that for something to be a subgroup, all group properties must hold
under the operation.
b)I guess this portion is telling us we have an abelian group since we
are essentially showing HK=KH.
HK=(x1,e)(e,x2)=(x1e,ex2)
KH=(e,x2)(x1,e)=(ex1,x2e)
G=(x1,x2)
Want to show (x1,e)(e,x2)=(e,x2)(x1,e)=(x1,x2)
We know HK=(x1,e)(e,x2)=(x1e,ex2). By properties of identity elements
e*x=x*e. So (x1e,ex2)=(ex1,x2e)=HK
Furthermore, e*x=x*e=x
Then HK=(x1,x2)=G

C)H=(x1,e), K=(e,x2)
We want H and K
So, (x1,e) and (e,x2)
I'm not sure how to go from there.

 

[micromass]

1) You've basically proved this in your last post.
2) Correct, but watch your notation: HK=(x1,e)(e,x2) is a notation which doesn't really makes sense to me. You'd have to write (x1,e)(e,x2)\in HK

3) Take an element in H. This has the form (x,e). What does it mean to say that this element also lies in K?

 

[kathrynag]

Ok, I realize 1) now.
For 3) let (x,e) be an element of H. If it also lies in K it must be of the form (e,x). So we have to have (e,e). Unsterstanding wise, I get this, but I don't think this would be a formal way of proving it and that's what I don't get.

 

[micromass]

No, it's a good proof. I don't see anything wrong with it...

 

[kathrynag]

Ok, I just wasn't sure if that was formal enough or not. I guess it shows everything that it should.