Verified answer

You haven't defined your notation -- and it's hard to read with _ and stuff.

What you might be asking is:

For σ in S[n], the permutation group on {1,2,...,n} and A⊆ {1,2,...,n} :

‹ σ › is transitive on A if and only if orb(a) = A for some a in A.

For G to be transitive on X means for any x,y in X, there is some g in G so that g(x)=y. Here orb(a) is the orbit of a under the subgroup ‹ σ ’. That subgroup is the cyclic group generated by σ.

Is this your question?

If so, here is the solution I would give:

Part 1 ( the ⇐ direction):

Assume orb(a) = A. Then choose b and c in A.

Since b and c are in A=orb(a) there are integers k and m such that:

      σ^k (a) = b and

      σ^m (a) = c

Then it should be relatively clear that:

      σ^(m-k) (b) = c

      σ^(k-m) (c) = b

which means that ‹ σ › is transitive on A.

Part 2 (the ⇒ direction):

Assume ‹ σ › acts transitively on A.

Then there is an element of ‹ σ › sending a to any other b in A. This means orb(a) = A by definition.

QED