DetermineK\triangleleft G,\ \frac{G}{K}\cong H_{1}andK\cong H_{2}

Nicontio1

Nicontio1

Answered question

2022-01-14

Determine
KG, GKH1
and
K=H2

Answer & Explanation

Tiefdruckot

Tiefdruckot

Beginner2022-01-15Added 46 answers

Step 1
G=H1×H2
Consider a map:
ϕ:H1×H2H1
defined by
ϕ(h1, h2)=h1
THen ϕ is a onto homomorhism (check!) from
H1×H2H1
ker(ϕ)={(h1,h2):ϕ(h1,h2)=eH1}
={(eH1,h):hH2}
=K
Since, ϕ:H1×H2H1
is an onto homomorphism with
ker(ϕ)=K
1) Hence, K is a normal subgroup of G=H1×H2
2) Then, by first Isomorhism Theorem, GK=H1
3) Define a map μ:KH2 by
μ(eH1,h)=h for all hH2
This gives an isomorphism form K to H2
Cheryl King

Cheryl King

Beginner2022-01-16Added 36 answers

Step 1
So for KG
Let
aH1, bH2
Then the conjugate of
(1,h)K
is
(a,b)(1,h)(a,b)1=(a1a1,bhb1)=(1,bhb1)
And you can probably make the final connection yourself.
Now for
GK=H1:
You want to find a homomorphism
ϕ:GH1
which is onto, such that
kerϕ=K
Think about a function that just pulls out the relevant 'coordinate' from (a,b)G and sends it to H1, and then just show that that it has the needed kernel and is a homomorphism. Then the first isomorphism theorem will produce the result.
Finally K=H2:
This is almost self-evident. Considering the elements in the two groups have the form (1,h) on the one hand and h on the other, the 'obvious' mapping between the two will be the isomorphism you need.

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