Prove that the alternating group $A_{n}$ is generated by all products of two 2-cycles ($i$,$j$)($k$,$l$). These 2-cycles are not necessarily disjoint.
So I know that $A_{n}$ can be written as products of even number of transpositions. In order to prove the statement, we have to prove that we can write any element in $A_{n}$ as ($i$,$j$)($k$,$l$). Equivalently, we have to prove that we can write any transposition using any product of two 2-cycles ($i$,$j$)($k$,$l$). Is my reasoning correct?
Best Answer
I think you have answered the question yourself. But your second sentence is wrong: "In order to prove the statement, we have to prove that we can write any element in $A_n$ as $(i,j)(k,l)$". That couldn't be true if $n$ were large. When we say that a group is generated by a set of elements, it doesn't mean that every element in the group is one of those; it means that every element can be written as a a finite number of products of them.