Given a finite group G, if G satisfies the property that any two maximal subgroups are conjugate, G is cyclic. I was thinking about Sylow theorem which states that for finite group and prime factor p, any Sylow p-subgroups are conjugate. I wish to somehow use that result to show G has order a prime number, thus cyclic. However, I am, at the moment, clueless. Any help or guide is greatly appreciated =).
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@David Hill, would you mind explaining further why is that ? – S_j Jun 25 '17 at 18:36
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@Thiet Look at the maximal subgroups of the group of quaternions. If you pick any pair, are they conjugate? – Code-Guru Jun 25 '17 at 18:47
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See this MO-question, and the solution in the comments. – Dietrich Burde Jun 25 '17 at 18:51