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Application of Sylow's Theorems

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a. Let G = Gl_3(F_p) denote the multiplicative group of all 3 x 3 nonsingular matrices with entries in the field of p elements F_p = Z/pZ. Show that the set of all matrices of the form

(1 a b
0 1 c
0 0 1)

where a,b, c belong to F_p, is a Sylow-p subgroup of G.

b. Is it the only Sylow-p subgroup of G?

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Solution Summary

In this solution, we use Sylow's theorems to show that a particular subset of G = GL(3,Z_p) is a Sylow subgroup of G.

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a. To show that the set H of matrices of the above form is a Sylow-p subgroup of G, it suffices to show that it is a subgroup of order p^k, where #G = p^k q with q not divisible by p. Clearly the order of H is p^3, since there are p choices for each of a, b, and c. We also know that the order of G is given by

#G = (p^3 - 1)(p^3 - p)(p^3 - p^2) = p^3(p^3 - 1)(p^2 - 1)(p - ...

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