where $T_n(F_p)$ denotes the the group of $n \times n$ invertible upper triangular matrices with entries in the field $F_p$ and $p$ is a prime. $O_3(F_2)$ is the orthogonal group. $SO_3(F_3)$ stands for the special orthogonal group. $SL_3(F_p)$ stands for the special linear group.
How to calculate the orders of these groups: $T_n(F_p) , SL_3(F_p) , O_3(F_2) , SO_3(F_3) , O_2(F_7)$
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group-theory
representation-theory
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0Well the last three are just specific groups. One could just write out the matrices and count. – 2011-11-16
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0For $SL_n$ see http://math.stackexchange.com/questions/71288/probability-of-having-a-determinant-of-1/71291#71291 – 2011-11-16
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0For $T_n$ start by looking at small examples and try to find a pattern. Hint the first orders of $T_n(\mathbb{F}_p)$ for a fixed prime $p$ are $p-1$ and $(p-1)^2p$ for n =1 and 2, respectively. – 2011-11-16
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0First off, you should make the question self-contained, *without* relying on the title. Second, issuing commands like this, with no reference to any work *you yourself* have done, is unlikely to elicit lots of answers. – 2012-02-20