Quoting"Let $H$ be a normal subgroup of $G$. Define the natural or canonical homomorphism. $$\phi : G \rightarrow G/H$$ by $$\phi(g) = gH$$ This is indeed a homomorphism, since $$\phi(g_1g_2) = g_1g_2H = g_1Hg_2H = \phi(g_1)\phi(g_2)$$ The kernel of this homomorphism is $H$."

I understand that: $$ker\phi = \phi^{-1}({e}) , e \in G/H$$.

I am not understanding 2 parts:

  • Part 1: How is it possible that $g_1g_2H = g_1Hg_2H $ ?
  • Part 2: How does the author come to the conclusion that $H$ is the Kernel?

Any input is much appreciated.


$\cdot$ Part 1: it is the operation of the group $G/H$.

$\cdot$ Part 2: $H$ is the identity of the group $G/H$. Then $$\ker\phi=\{g\in G:gH=H\}=\{g\in G:g\in H\}=H.$$


Also to define the quotient group $G/H$ you must have that $H$ is a normal subgroup of $G$ in order that the operation in $G/H$ to be well defined.

Thus you have $g_1g_2H=g_1Hg_2H$ when $H$ is a normal subgroup of $G$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.