# Subgroup of $S_4$ generated by $\{(123), (12)(34)\}$

by wittbluenote   Last Updated April 15, 2019 11:20 AM - source

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!

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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 $$S$$.