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.