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I'm interested in the following problem from Dummit & Foote's Abstract Algebra text (Exercise 40 of Section 4.5):

Prove that the number of Sylow p-subgroups of $GL_2(\mathbb{F}_p)$ is $p+1$. [Exhibit two distinct Sylow p-subgroups.]


I've managed to solve the exercise in the following way:

Exercise 39 tells us that $UT_2(\mathbb{F_p})$, the subgroup of upper triangular matrices with ones on the diagonal is a p-Sylow subgroup. It can be verified that the normalizer $N_{GL_2(\mathbb{F}_p)}(UT_2(\mathbb{F}_p)$ is a supergroup of the larger subgroup of upper triangular matrices, where the diagonal entries are allowed to be any nonzero element of the field $\mathbb{F}_p$ (this subgroup has size $p(p-1)^2$). Since $$n_p=|GL_2(\mathbb{F}_p):N_{GL_2(\mathbb{F}_p)}(UT_2(\mathbb{F}_p))|,$$ the above gives $$n_p=\frac{p+1}{m}$$ for some integer $m$. Part (iii) of Sylow's Theorem then forces $n_p=p+1$. (The case $n_p=1$ is impossible, as the subgroup of lower triangular matrices is a Sylow p-subgroup as well.)


I think my solution is correct (isn't it?), however, I wonder, do simpler solutions exist? Thank you!

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    $\begingroup$ Yes. Your solution is correct. $\endgroup$ – Jyrki Lahtonen Dec 14 '14 at 21:34

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