Let $p$ be prime divisor of order of finite group $G$, and the number of cyclic subgroup of order $p$ be $p+1$. If $P$ is a Sylow $p$-subgroup of $G$, then $P$ is normal in $G$ and $|P|=p^{2}$($P$ is not cyclic). Also let the number of cyclic subgroup of order $2$ be $p(p+1)/2$. Is it true the number of cyclic subgroup of order $2p$ is a multiple of $p+1$?
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I believe the semidirect product of an elementary abelian group of order 25 with a dihedral group of order 6 and faithful action is a counterexample. This is SmallGroup(150,5) in the GAP or Magma library. There are 6 subgroups of order 5, 15 of order 2 and 15 cyclic subgroups of order 10. Note that 15 is not a multiple of 6. More generally, if $q = (p+1)/2$ is odd, then I would expect the semidirect product of elementary $p^2$ with dihedral $2q$ (and faithful action) to be a counterexample. You would have a better chance if you asked whether $t_{2p}$ was a multiple of $(p+1)/2$. |
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