Exercise 3.5.15 ($A_4 = \langle x, y angle$, where $x,y$ are distinct $3$-cycles, $x
e y^{-1}$.)

Question

Prove that if $x$ and $y$ are distinct $3$-cycles in $S_4$ with $x 
e y^{-1}$, then the subgroup of $S_4$ generated by $x$ and $y$ is $A_4$.

Richard Ganaye · Accepted Answer

\begin{proof} 
Let $x$ and $y$ be distinct $3$-cycles in $S_4$ with $x 
e y^{-1}$. Then $x \in A_4,\ y \in A_4$. Set $H = \langle x, y angle$.  The supports of $x$ and $y$ have two elements in common, say $a,b$, but not three, otherwise $x = y$ or $x = y^{-1}$. Since $\{x, y ,x^{-1}, y^{-1}\} \subset A_4$,
$$(a\ b \ c) \in H\qquad 	ext{and} \qquad (a \ b \ d) \in H,$$
where $a,b,c,d$ are distinct.
Then 
$$(a\ b \ c) (a \ b \ d) = (a \ c) (b \ d)$$
has order $2$, so $H$ contains an element of order $2$ and an element of order 3. By Exercise 14, 
$$A_4 = \langle x, y angle.$$

\end{proof}

Exercise 3.5.15 ($A_4 = \langle x, y \rangle$, where $x,y$ are distinct $3$-cycles, $x \ne y^{-1}$.)

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Exercise 3.5.15 ($A_4 = \langle x, y \rangle$, where $x,y$ are distinct $3$-cycles, $x \ne y^{-1}$.)

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