I refer to the following problem.
Determine the subgroup of $S_4$ generated by $\{(123), (12)(34)\}$.
In his solution to the problem the author makes the following claim:
As $(123) \in A_4$ and $ (12)(34) \in A_4$, then certainly $ \langle S \rangle \leq A_4$.
It is this claim that is of concern to me (not the above problem). In particular, I fail to see how the implication
$(123) \in A_4$ and $ (12)(34) \in A_4 \implies \ \langle S \rangle \ \leq A_4$
is immediate or self evident.
What am I missing? Any help would be greatly appreciated!
Well, $\langle S \rangle$ is defined as the smallest subgroup (of a given group $G$) which contains $S \subseteq G$. Now there is no doubt that $A_{4}$ is a subgroup of $G = S_{4}$, and that $A_{4}$ contains $S$.