Sylow subgroup related

Question

In a group of all invertible $4 \times 4$ matrices with enteries in the field of $3$ elements, What is the cardinality of any $3$- sylow subgroup?

Have you found the order of the group? – Tobias Kildetoft Jun 22 '16 at 06:43 — Tobias Kildetoft, Jun 22 '16 at 06:43

score 1 · Answer 1 · answered Jun 22 '16 at 07:43

In addition to, and to explain the very brief answer of @kalpeshmpopat, I advice you to study a bit of General Linear groups, that is, all invertible $n \times n$ matrices over a finite field $\mathbb{F}_q$, denoted by $GL(n,q)$. It is not hard to show that $$|GL(n,q)|=\Pi_{i=0}^{n-1}(q^n - q^i)=q^{n(n-1)/2}\Pi_{i=1}^{n}(q^i -1).$$ So if $q=p^r$, the Sylow $p$-subgroup of $GL(n,q)$ has order $q^{n(n-1)/2}$. The subgroup of upper triangular matrices with $1$’s along the diagonal is a Sylow $p$-subgroup, which can be seen by counting the possibilities for the coefficients of these kind of matrices. You can replace the word "upper" here by "lower" to obtain conjugates of the Sylow subgroup.

score 0 · Accepted Answer · answered Jun 22 '16 at 06:44

0

It is given by formula $p^{\Sigma(n-1)}$.

Hence in this case its is $3^{\Sigma(4-1)}=3^6=729$

answered Jun 22 '16 at 06:44

kalpeshmpopat

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Sylow subgroup related

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