Isomorphism

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Joined: Jan 12, 2011

by iarnab_kundu, Nov 13, 2013, 8:24 AM

Result - Let $G$ be a group with subgroups $H$ and $K$ such that $HK$ is also a subgroup. Then we have that $HK/H\cong K/H\cap K$ .

Let's analyze the sets $HK=\{hk|h\in H\ \text{and}\ k\in K\}=\bigcup_{k\in K}kH$ .
Where as $HK/H=\{(kH)H| k\in K\}=\{kH|k\in K\}$ .

Its pretty well known that $HK/H$ is a group.
Let's take an homomorphism $f:K\to HK/H$ , such that for any $k\in K$ $f(k)=kH$ Let's try to find its kernel.

Suppose $kH=H\iff k\in H\iff k\in H\cap K$ . Overtly the kernel is $H\cup K$ .
This forces an isomorphism form $K/H\cap K$ to $HK/H$ .
$\blacksquare$ .

Corollary - Let $G$ be a group with subgroup $H$ and normal subgroup $N$ . Then $NH/H\cong N/N\cap H$ .

This post has been edited 4 times. Last edited by iarnab_kundu, Nov 29, 2013, 6:09 PM

Cracking the Nut Open

Isomorphism

by iarnab_kundu, Nov 13, 2013, 8:24 AM

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