kernel, group, and subgroup…

Question

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.

Answer 1

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

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

Answer 2

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 $G$