Math Help - Group Theory Tough Problems

1. Group Theory Tough Problems

Howdy all, having issues with some group theory problems. Hints or suggestions are very welcomed.

(1) Prove that for n>=3 the subgroup generated by the three cycles is An.

(2) Prove that the smallest subgroup of Sn containing (1, 2) and (1, 2, . . . , n) is Sn

This is what I have so far:

Sn is generated by 2-cycles. Any element in An is a product of an even number of transpositions, so we only need to show the product of any two transpositions can be written in terms of 3-cycles.
(12)(13) = (123), while
(12)(34) = (123)(143).

Is this sufficient enough, I feel as if this is not proofy. HELP!!

2. Originally Posted by RoboMyster5
Howdy all, having issues with some group theory problems. Hints or suggestions are very welcomed.

(1) Prove that for n>=3 the subgroup generated by the three cycles is An.

Note that any permutation writable as a 3-cycle is an even permutation thus clearly the subgroup generated by all three cycles is a subgroup of $A_n$. To show the other inclusion we first note that any $\sigma\in A_n$ may be written as the product of an even number of transpositions. It suffices then to show that the product of two distinct transpositions can be written as a 3-cycle. To do this we break it into two cases:

1. The transpositions aren't disjoin, in other words we have $(a,b)(a,c)$. But a little work shows that the above is actually equal to $(a,c,b)$

2. The transpositions are disjoint, or we have $(a,b)(c,d)$. Again, with a little work, we can show that this is equivalent to $(a,b,d)(c,b,d)$

This finishes the proof.

What have you done for the second one?

3. See I'm not sure where to begin. I Know the symmetric group is generated bythe set of all transpositions. Since every permutation can be written as a product of disjoint cycles and each cycle is a product of transpositions. And I beleive that those two cycles are generators of the group. Thats all I can fathom.

4. Originally Posted by RoboMyster5
See I'm not sure where to begin. I Know the symmetric group is generated bythe set of all transpositions. Since every permutation can be written as a product of disjoint cycles and each cycle is a product of transpositions. And I beleive that those two cycles are generators of the group. Thats all I can fathom.
But that is all you need! Think about it is like this, suppose you have some group $G$ such that $(1,2),(1,\cdots,n)\in G$, if you can prove that $(1,2),\left(1,\cdots,n\right)$ generates $S_{n}$ then clearly $S_n\le G$ but remeber here that we are only considering $G\le S_n$. Put the two together and we get $G=S_n$!