Exercise 3.5.17 (Isomorphism type of $\langle x, y \rangle \leq S_n$, where $x$ and $y$ are $3$-cycles)

Question

If $x$ and $y$ are $3$-cycles in $S_n$, prove that $\langle x, y angle$ is isomorphic to $Z_3,\ A_4, A_5$ or $Z_3 	imes Z_3$.

Richard Ganaye · Accepted Answer

\begin{proof} 
If $x$ and $y$ are disjoint $3$-cycles in $S_n$, then $x$ and $y$ commute, therefore
$$\langle x, y angle \simeq  \langle x angle 	imes \langle y angle,$$
so
$$\langle x, y angle \simeq Z_3 	imes Z_3.$$

If $x$ and $y$ are not disjoint, the union of their  two supports has at most $5$ elements, so  up to conjugation we may suppose that $\langle x,y angle \leq  S_5 \leq S_n$.

By Exercise 16, if $x 
e y$ and $x 
e y^{-1}$, then 
$$\langle x, y angle \simeq A_4\qquad 	ext{or} \qquad \langle x, y angle \simeq A_5.$$

If $x= y$, or $x = y^{-1}$, then
$$\langle x, y angle = \langle x angle \simeq Z_3.$$
In conclusion, if $x$ and $y$ are $3$-cycles in $S_n$, then $\langle x, y angle$ is isomorphic to $Z_3,\ A_4, A_5$ or $Z_3 	imes Z_3$.
\end{proof}

Exercise 3.5.17 (Isomorphism type of $\langle x, y \rangle \leq S_n$, where $x$ and $y$ are $3$-cycles)

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Exercise 3.5.17 (Isomorphism type of $\langle x, y \rangle \leq S_n$, where $x$ and $y$ are $3$-cycles)

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