Question

Prove that for n >=3 a n is generated by the 3-cycles

Answer 1

Answer:

For my course in Group Theory, I have seen various proofs that show why the alternating group An, which consists of the elements of Sn that can be expressed as an even number of transpositions (i.e. 2-cycles), is generated by the 3-cycles.

All of these proofs, and sometimes also the question, seem to guide you to showing that any element in An can be expressed as a product of 3-cycles. Now I get the proofs up to this point.

What I do not understand, and I hope you can help me with, is why the fact that any element in An can be expressed as a product of 3-cycles means that An is generated by the 3-cycles. Could it not be that, even though any element of An can be expressed as a product of 3-cycles, that if we let the 3-cycles generate a group there will be elements in that group that are not in An? I do not see why our proof (for instance given here) would exclude that possibility.

If any of you could shed some light on this, your help is very much appreciated!