# On the $PSL(2, p)$, $p$ a Mersenne prime

We know $|PSL(2,p)|=p(p+1)(p-1)/2$. Let $p$ be Mersenne prime (that is $p+1=2^{n}$) and $r$ be prime divisor of $(p-1)/2$.

My question: What is the number of Sylow $r$-subgroups of $PSL(2,p)$?

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Let $G=\operatorname{PSL}(2,q)$ for q an odd prime power. If r is an odd prime not dividing q but dividing $|G|$, then r divides exactly one of $\{q+1,q-1\}$ and a Sylow r-subgroup of G is cyclic with normalizer dihedral of order $q-\epsilon$ ($\epsilon = \pm1$). In particular, the number of Sylow r-subgroups is $q(q+\epsilon)/2$.
This does not require q to be a Mersenne prime, or even prime (just odd), and does not require r divide $q-1$ (though in your case this means $\epsilon=1$).
I was about to start typing an answer based on a calculation showing that that the normalizer of a non-trivial subgroup of the subgroup of the diagonal matrices (such as these subgroups of order $\mid (p-1)$) consists of the monomial matrices, and thus is dihedral of order $2(p-1)/2$. +1 and thanks for saving me the trouble :-) – Jyrki Lahtonen Jun 12 '12 at 16:02