I'm a bit stuck on this question, any help will be appreciated, here's the question:
Let $G$ be a group with an element $g \in G$ such that $o(g) = 2$ and assume $g \in Z(G)$. Let $K$ be a subset of $G \times \mathbb{Z}_2$ defined by $K = \{(e, 0), (g, 1)\}$. Is it true that $(G \times \mathbb{Z}_2)/K \cong G$?
Here is my attempt:
I was trying to use First Isomorphism Theorem, so I need to find a homomorphism map such that it's kernel is $K$ and it's image is $G$, but I'm having difficulties to find such map.
Since it has $(g,1)$ in it's kernel, which means the map should have something squared in this mapping, but the binary operation from $G$ is not guaranteed to be commutative, so I'm stuck, i.e.,
$$(a \ast x )^2 \neq a^2 \ast x ^2$$
If there really does not exist such a map, how do I even prove this statement is false?
A general result is that if $H$ and $K$ are normal subgroups of a group $G$ with $HK=G$ and $H\cap K=\{e\}$, then $G\cong H\times K$. (1. $hk=kh$ since $hkh^{1}k^{-1}\in H\cap K$. 2. the map $H\times K\to G$, $(h,k)\mapsto hk$ is surjective with kernel $H\cap K$.)
In the present example, take $H=G\times\{0\}\cong G$. Then, $G\times\mathbb{Z}_2=HK$ and clearly $H\cap K=\{(e,0)\}$. Therefore, $G\times \mathbb{Z}_2\cong G\times K$.
Finally, in general, $(G\times K)/K\cong G$ (the projection $G\times K\to G$, $(g,k)\mapsto g$ is a surjective homomorphism with kernel $K$).