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.
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Well the last three are just specific groups. One could just write out the matrices and count. – jspecter Nov 16 '11 at 05:27
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For $SL_n$ see http://math.stackexchange.com/questions/71288/probability-of-having-a-determinant-of-1/71291#71291 – jspecter Nov 16 '11 at 05:28
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For $T_n$ 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. – jspecter Nov 16 '11 at 05:31
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First 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. – Feb 20 '12 at 22:26
