For $n>1$ ;
Let $H$ be the set of all permutations in $S_n$ that can be expressed as a product of a multiple of $4$ transpositions .Show that $H=A_n$
We know that $A_n$ is generated by $3-$ cycles .
Now $H\subset A_4$ since every element of $H$ is a product of even number of transpositions.
Now take any $\alpha\in A_n$ then $\alpha$ is a $3$-cycle or a product of $3$ cycles.
If $\alpha$ is a product of $3$ cycles of the form $\alpha=(x_1x_2x_3)\ldots (y_1y_2y_3)$ then $\alpha=(x_1x_3)(x_2x_3)\ldots (y_1y_3)(y_1y_2)\in H$.
If $\alpha$ is a $3-$ cycle then $\alpha=(x_1x_2x_3)$
If $\alpha\in A_3$ and if we suppose $\alpha=(123)$ in $A_3$ then how can we write $\alpha$ as a product of $4$ $2-$ cycles?