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Let $G$ be a finite non-cyclic group such that it has cyclic subgroup of order $n$. Please consider the following claim:

The number of cyclic subgroups of order $n$ in $G$ is a multiple of the greatest divisor of $|G|$ that is prime to $n$.

I have checked it for many finite groups and think must be true. But I cannot prove it. What is your idea? Thanks

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  • $\begingroup$ $C_6$ has unique cyclic subgroups of orders $2$ and $3$. But $1$ is not a multiple of the greatest divisor of $|G|$ that is prime to $n$. Or did you mean to write FACTOR instead of MULTIPLE? $\endgroup$
    – Nick Gill
    Mar 29, 2013 at 15:37
  • $\begingroup$ @Nick Gill: I consider only non-cyclic group. $\endgroup$
    – U-samir
    Mar 29, 2013 at 16:01

1 Answer 1

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A counterexample to your claim: The symmetric group $S_3$ contains a unique cyclic subgroup of order 3 (the alternating group $A_3$), however 1 is not a multiple of 2 (the greatest divisor of $|S_3|=6$ that is relative prime to 3).

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