Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Is there a way to classify all $p$-Sylow subgroups of $GL(n,\, \mathbb{F}_{p^k})$ where $|\mathbb{F}_{p^k}|=p^k$? Clearly the prime $p$ (that is the characteristic of the base field) is a very privileged number; for example we can say that a $p$-Sylow of $GL(n,\, \mathbb{F}_{p^k})$ has order $p^{k(1+2+\ldots+(n-1))}$. But what about the structure of these $p$-Sylows can we identify them as normalizers/centralizers of flags in the projective geometry?

share|cite|improve this question

The upper triangular matrices with diagonal entries $1$ form a $p$-Sylow subgroup $P$. So the set of all $p$-Sylow subgroup is given by all the conjugates of $P$.

EDIT: In the light of the answer of Andreas Caranti, $P$ is contained in the stabilizer of the flag $$\{0\} < \langle e_n\rangle < \langle e_{n-1}, e_n\rangle < \ldots < \langle e_2, e_3,\ldots, e_n\rangle < \mathbb{F}_{p^k}^n,$$ where $e_i$ denotes the $i$-th standard vector.

EDIT 2, thanks to the comment of Jyrki Lahtonen: The full stabilizer is still a bit larger than $P$. It is the set of all invertible upper triangular matrices.

share|cite|improve this answer
Won't all the upper triangular matrices stabilize this flag? Irrespective of whether they have ones on the diagonal or not? IOW aren't the Borel subgroups the stabilizers of maximal flags. – Jyrki Lahtonen Mar 12 '13 at 21:55
@JyrkiLahtonen: Thanks for pointing this out. Your are right. – azimut Mar 12 '13 at 21:58
My problem is that the above (complete) flag is not a flag of the projective geometry associated to the vector space $V$ because it contains $\{0\}$ and $V$. The elements of $PG(V)$ are only proper nonzero subspaces. – Dubious Mar 12 '13 at 21:58
This doesn't change anything. $\{0\}$ and $V$ are stabilized by any invertible linear map. So you can safely drop them. – azimut Mar 12 '13 at 22:01

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.