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Jane,
Wish it were that easy. Take G of order 7*5*3. n(7) = 15 is both 1 (mod 7) and divides 5*3. And n(5) could be 21; and n(3) could be 7.
So there is no guaranteed uniquess of the Sylow subgp for any of the prime factors.
Any other ideas?
Jim
G has order pqr (p > q> r). It's easy to use Sylow Theorems directly to show G is not simple (just a counting argument based on n(p), n(q), n(r) -- the numbers of Sylow p-, q-, and r-subgroups).
But, can we say more in this case. E.g., I would suspect that the Syl p-group is unique. Perhaps the Syl q-gp as well.
Any ideas?
Jim
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