Math Problem Statement

Let G be a group of order 45.

  1. Determine the number of Sylow 3-subgroups of G.
  2. Let H ⊆ G be a Sylow 3-subgroup of G. a. Show that H has an element of order 3. b. Use the above element to show that H cannot be simple. (Hint: Start by concluding that H is abelian.)
  3. Prove that G has a subgroup that is isomorphic to Z/5Z. (Note that this can be solved independently from 2.)

Solution

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Math Problem Analysis

Mathematical Concepts

Group Theory
Sylow Theorems
Abelian Groups
Isomorphism

Formulas

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Theorems

Sylow theorems
Cauchy's theorem

Suitable Grade Level

Advanced Undergraduate