Purchase Solution

Application of Sylow's Theorems

Not what you're looking for?

Ask Custom Question

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?

Purchase this Solution
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.

Solution Preview

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 - ...

Purchase this Solution
Free BrainMass Quizzes
Probability Quiz

Some questions on probability

Graphs and Functions

This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.

Exponential Expressions

In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.

Solving quadratic inequalities

This quiz test you on how well you are familiar with solving quadratic inequalities.

Multiplying Complex Numbers

This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.

View More Free Quizzes
Related BrainMass Solutions

  • Proof that There Are No Simple Groups of Order 56 or 148

    In this solution, we use Sylow's theorems to prove that no simple groups exist of order 56 or 148.

  • Sylow's theorem

    363918 Proof: Generalization of the last Sylow's theorem Prove the following generalization of the last Sylow's theorem: If |G| is divisible by p^b, and H <= G has order p^a with a <= b, then the number of subgroups of G that both contain H and have order

  • Sylow's Theorem

    Modern Algebra Group Theory (CIX) Sylow's Theorem In the symmetric group of degree 4, S4 , find a 2-Sylow subgroup and a 3-Sylow subgroup.

  • Sylow's Theorem

    Modern Algebra Group Theory (CX) Sylow's Theorem Find all 3-Sylow subgroups of or, Sylow 3-subgroups and 2-Sylow subgroup or, Sylow 2-subgroups of the

  • Using Divergence (Gauss) and Curl (Stokes) Theorems

    The point of this assignment is that you reproduce the application of the Divergence (Gauss) and Curl (Stokes) theorems in a way similar to what you have been shown in your textbook and/or in class.

  • Sylow's Theorem

    387631 Clarification of Sylow's Theorem A) Prove there is no simple group of order 200. b) Assume that a group G has two Sylow p-subgroups K and H. Prove that K and H are isomorphic.

  • Sylow's Theorem Problems

    519325 Sylow's Theorem Problems Verify that if H is a subgroup of G, and a ? G, then aHa^-1 is a subgroup of G. Prove that if H is a finite subgroup of G, and a ? G, then |aHa^-1| = |H|. (Suggestion: The mapping h ? aha^-1 is one-to-one.)

  • Application of Stokes Theorem

    Thus: And the curve looks like this (standard view and top view): The expert examines the application of Stokes theorems.

  • Subgroup proofsno

    Suppose has Sylow -subgroups, then according to the Third Sylow's Theorem, and (mod . So . Since , we must have and hence . So is the unique Sylow -subgroup of . Also from the third Sylow's Theorem, is normal in .

View More Related Solutions