conditions on m-cycles generating a symmetric group S_n

Hi,

I have a question on sufficient conditions for m-cycles generating a
symmetric group S_n (n>4):
Given is a set C of m-cycles, i.e. cycles
of fixed length m, of the symmetric group S_n (n>4, m<n), where m is a
positive even integer. C has the following two properties:
1. The union of the supports of the m-cycles of C is equal to the set
1,2,3,..,n
2. ("connectedness") If T is a proper
subset of C, then the union of the supports of the m-cycles of T has a
non-empty intersection with the union of the supports of the m-cycles
of C-T. (So #C>=(n-1)/(m-1)).

My question: Do the elements of C generate S_n? If not, what additional
(non-trivial) property is necessary?
Example: n=13, m=4: Do (1 2 3 4) (4 5 6 7) (7 8 9 10) (10 11 12
13)generate S_13?

What I already see is that the two above mentioned conditions are
necessary for C being minimal as to the generation of S_n.This is due
to the fact that the only non-trivial normal subgroup of the symmetric
group S_n (n>4) is the subgroup A_n of alternating permutations.


Thanks a lot for your advise.
Adrian

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