# Question on generating sets of $S_n$

This is the question:

Decide whether the following sets generate $$S_n$$ or not:

1. The set of 2-cycles in $$S_n$$

2. The set of even permutations in $$S_n$$

3. The set of odd permutations in $$S_n$$

4. The set of 3-cycles in $$S_n$$

Here's my justifications and attempts:

(1) is true since every permutation in $$S_n$$ can be written as a product of 2-cycles.

(2) is false since the set of even permutations is a subgroup of $$S_n$$ and it is closed. Thus, none of the odd permutation can be written as products of even permutations.

(3) The set of odd permutation contains the set of 2-cycles which generate $$S_n$$, thus, the set of odd permutation generates $$S_n$$.

I'm not sure what to conclude for (4). However, I can conclude that the set of 3-cycles in $$S_n$$ generates $$A_n$$ for every even permutation can be written as a product of 3-cycles using the fact that $$(ab)(bc)=(abc)$$ and $$(ab)(cd)=(cba)(acd)$$ but does that prevent it from generating $$S_n$$? I need some hints.

The 3-cycle set indeed only generates $$A_n$$, not $$S_n$$. 3-cycles and their compositions/inverses are even permutations, but $$S_n$$ contains odd permutations.