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There are restrictions on the values of a cyclotomic polynomial evaluated at an integer that are reminiscent of the restrictions on the number of Sylow groups of a group. So I'd like to know if there is a connection.

Does the value of a cyclotomic polynomial evaluated at an integer count something related to the properties of a certain group (for example, something related to its Sylow groups)?

For example, if $n_p$ is the number of $p$-Sylow groups of the group $G$ with $|G|=n=p^\alpha r$, then $n_p\equiv 1 \pmod p$. On the other hand, if $p | \Phi_n(x)$ and $p \not| n$, then $p\equiv 1 \pmod n$. Since the cyclotomic polynomial are closely related to the cyclic groups, their values might count something about the group.

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Could you please give examples of such restrictions and some references? –  lhf Nov 21 '12 at 20:36

1 Answer 1

I believe $$ \Phi_n(r) = \frac{LCM(r^k-1, k=1\ldots n)}{LCM(r^k-1,k=1\ldots n-1)} $$ for integers $n \ge 1$ and $r \ge 2$.

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How is that related to groups? –  lhf Nov 22 '12 at 18:21
    
Maybe it isn't, but I thought this formulation might be helpful. LCM really belongs to ring theory rather than group theory. –  Robert Israel Nov 22 '12 at 22:12
    
I didn't say it wasn't. I was (am) curious if it is. –  lhf Nov 22 '12 at 22:14

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