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Jösus
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Homework Statement
I am trying to prove that the alternating group on five letters, A5, contains no subgroups of order 20.
Homework Equations
I guess nothing is needed here, for this problem. Though I will use this extra space to explain my notation, if it would happen to differ from the standard one:
Sym(5) - The symmetric group on five letters.
A5 - The alternating group on five letters (The subgroup of Sym(5) consisting of all even permutations).
The Attempt at a Solution
I've tried to solve it for some time now, but none of my attemps have been fruitful at all. My first idea was to prove that the subgroup H of Sym(5) generated by a 5-cycle and a 4-cycle was, at least up to isomorphism, the only subgroup of Sym(5) having order 20. As the 4-cycle can be written as a product of 3 transpositions, it would then follow that the cycle is an odd permutation, and hence not in A5. However, I seem to be unable to prove this result.
Another intuition I had was that, maybe, if d divides the order of a subgroup H of Sym(n) and d is smaller than or equal to n, then H must contain all d-cycles. If that were to be true then any such subgroup of order 20 would have to contain all 4-cycles, which are odd. Though, this too I seem to be unable to prove. As a matter of fact, I have no longer any idea if it is true or not.
Anyway, I would appreciate some help with this, if possible.
Thanks in advance