- #1
kathrynag
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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.