Is the $S_4\times G$ solvable group?

Question

We have the following claim : The group $G$ is solvable iff $S_4\times G$ is solvable. If we consider that $S_4\times G$ is solvable we have that $1\times G\leq S_4\times G$ is solvable as a subgroup of solvable group.We consider the isomorphism $$f:\ 1\times G \to G ,\ (1,g)\mapsto g\ ,$$ so we conclude that G is solvable. From the other hand if we consider that G is solvable we khow that exists a sequense : $$1=G_m \vartriangleleft G_{m-1} \vartriangleleft \cdots \vartriangleleft G_0=G$$ and $G_{n-1}/G_{n}$ is abelian group, $\forall n\in \{1,\cdots, m\}$.Also we khow that $S_4$ is solvable with the following sequence $$1\vartriangleleft H \vartriangleleft A_4 \vartriangleleft S_4$$ with $H=\{1,(12)(34),(13)(24),(14)(23)\}$.How can we show that $S_4\times G$ is solvable group?

Answer 1

Direct product of any two solvable groups is solvable. More generally any extension of a solvable group by a solvable group is solvable.