In $(G_1\times G_2)/G_2$, I am confused since $G_2$ is clearly not a subgroup of $G_1\times G_2$

Question

I have seen the following expression in the text book of algebra chapter$0$. $(G_1\times G_2)/G_2$. I am confused since $G_2$ is clearly not a subgroup of $G_1\times G_2$, and hence not a normal subgroup. Is it still valid to write this expression when $G_2$ is not a normal subgroup of $G_1\times G_2$?

Answer 1

It means

$$(G_1\times G_2)/(\{e_{G_1}\}\times G_2),$$

where $\{e_{G_1}\}$ is understood to mean the trivial subgroup of $G_1$ and

$$\{e_{G_1}\}\times G_2=\{(e_{G_1},g_2)\mid g_2\in G_2\}$$

is the normal subgroup of $G_1\times G_2$ (viewed, in a sense, as an internal direct product of $\{e_{G_1}\}$ and $G_2$).

As said in the comments, this is an abuse of notation. It works conceptually (most of the time) as $G_2$ is isomorphic to $\{e_{G_1}\}\times G_2$.

WARNING:

In $G=\Bbb Z_2\times \Bbb Z_2$, there are three subgroups of $G$ isomorphic to $\Bbb Z_2$, each of which giving a different quotient, rendering

$$\color{red}{(\Bbb Z_2\times \Bbb Z_2)/\Bbb Z_2}$$

ambiguous. Many such ambiguities in other quotients exist.