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!

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 $S$.

April 15, 2019 10:50 AM

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