Exercise 4.6.4 ($A_n$ is generated by the set of $3$-cycles)

Question

Prove that $A_n$ is generated by the set of all $3$-cycles for each $n \geq 3$.

Richard Ganaye · Accepted Answer

\begin{proof} 
Let $\mathscr{C}_3$ denote the set of $3$-cycles. We prove
$$A_5 = \langle \mathscr{C}_3 angle$$
(i.e., $A_5$ is generated by the $3$-cycles.)

Every element $f$ of $A_n$ is a product of an even number of transpositions $t_i$:
\begin{align*}
f &= t_1 t_2 \cdots t_{2k-1} t_{2k}\
&= \varphi_1 \varphi_2 \cdots \varphi_k,
\end{align*}
where each factor $\varphi_i = t_{2i} t_{2i}$ is a product of two transpositions. It remains to be proven that such a product
$$\varphi = (a\ b)(c\ d)$$
is itself a product of 3-cycles.

\begin{itemize}
\item If $\{a,b\} = \{c,d\}$, then $\varphi = (a\ b)(a\ b) = () = 1$.

\item If $|\{a,b\} \cap \{c,d\}| = 1$, we can assume $b = c$, since $(a\ b) = (b\ a)$. Then
$$\varphi = (a\ b)(b\ d) = (a\ b\ d)$$
is a 3-cycle.

If $\{a,b\} \cap \{c,d\} = \varnothing$, then
\begin{align*}
\varphi &= (a\ b)(c\ d)\
&= (a\ b) (b\ c) (b\ c) (c\ d)\
&= (a\ b \ c)(b\ c\ d)
\end{align*}
is the product of two $3$ cycles.

\end{itemize}
This proves $A_5 = \langle \mathscr{C}_3 angle$.
\end{proof}

Exercise 4.6.4 ($A_n$ is generated by the set of $3$-cycles)

Answers

Comments

Exercise 4.6.4 ($A_n$ is generated by the set of $3$-cycles)

Answers

Comments

Add answer