Groups of Order pq (Pt. II)
Point of Post: This is a continuation of this post.
Case 3 and
Lemma 1: Let be a prime, then .\ (where is the group of units of the field )
Proof: Define by . This is evidently a homomorphism since it’s injective since is a generator of and it’s surjective because given any one has that is an automorphism of .
Remark: Note there was no need for to be prime here, the above proof shows that for any one has that .
Thus, it remains to prove that . To do this (since ) it suffices to prove that is cyclic. To do this we note that if there does not exist an element of of order then there exists some such that for every . But, this implies that the polynomial has more solutions than its degree which is impossible. The conclusion follows.
Remark: The above proof that is cyclic is a little wishywashy. A more rigorous, and more general proof can be found in this post of mine.
Lemma 2: Let be a homomorphism. Suppose there are maps and such that (where is the inner autormorphism induced by in ). Then, .
Proof: This is mostly just grunt work. Namely, define by . As indicated, is an isomorphism. Indeed, to see it’s a homomorphism we note that
But, evidently is a bijection since a quick check shows that serves as an inverse.
Lemma 3: If is cyclic and are such that is conjugate to then .
Proof: By assumption for some and there exists some such that . Thus, for some generator of the cyclic . Since is a generator we know that and we may evidently then assume that . We can then define and . But, a quick check shows that and so the lemma follows from lemma 2.
We are now prepared to prove our main theorem. Namely:
Theorem: Let and be primes such that . Then, there exists a unique, up to isomorphism, nonabelian group of order .
Proof: It follows by our previous argument that there is precisely one Sylow subgroup of and Sylow subgroups. If represents the latter and one of the former then we have that is trivial, , and . Thus, for some homomorphism . Thus, we desire to seek how many of these homomorphisms define nonisomorphic nonabelian structures . But, since and this evidently is equivalent to finding the number of nonisomorphic nonabelian for some homomorphism . But, by our first lemma we know that and so evidently every nontrivial must have image with order . But, since is cyclic there is precisely one subgroup of order . Thus, for every nontrivial one has that and so by lemma 3 we know that . The conclusion follows.
Thus, every group with where are primes with is either isomorphic to or where is any nontrivial (and since there are evidently of them) homomorphism.
References:
1. Rotman, Joseph J. Advanced Modern Algebra. Providence, RI: American Mathematical Society, 2010. Print
2. Simon, Barry. Representations of Finite and Compact Groups. Providence, RI: American Math. Soc., 1996. Print
April 19, 2011  Posted by Alex Youcis  Algebra, Group Theory, Representation Theory  Algebra, Classifying, Group Theory, Groups of order pq, Representation Theory
3 Comments »
Leave a Reply Cancel reply
About me
My name is Alex Youcis. I am currently a senior a first year graduate student at the University of California, Berkeley.
About Me
Blogroll
 Absolutely Useless
 Abstract Algebra
 Annoying Precision
 Aquazorcarson's Blog
 Bruno's Math Blog
 Chris' Math Blog
 Climbing Mount Bourbaki
 E. Kowalski's Blog
 Geometric Group Theory
 Geometry and the Imagination
 Gower's Weblog
 Hard Arithmetic
 HardyRamanujan Letters
 Ngô Quốc Anh's Blog
 Project Crazy Project
 Rigorous Trivialities
 Secret Blogging Seminar
 SymOmega
 TCS Math
 Unapologetic Mathematician
 What's New
Lecture Notes
Categories
Top Posts
 Munkres Chapter 2 Section 17
 Munkres Chapter two Section 12 & 13: Topological Spaces and Bases
 Halmos Chapter one Sections 15, 16 and 17: Dual Bases, Reflexivity, and Annihilators (Part II)
 The Dimension of R over Q
 Endomorphism Ring of an Abelian Group
 Halmos Chapter one Sections 15, 16 and 17: Dual Bases, Reflexivity, and Annihilators (Part I)
 About me
 The Fundamental Theorem of Plane Curve Curvature
 Category Theory: Monoids Part II: Monoid Homomorphisms and Submonoids
 Munkres Chapter 2 Section 18

Algebra Algebraic Combinatorics Algebraic Topology Analysis Answers Category Theory Chapter 2 Characters Character Theory Class Functions Compactness Complex Analysis Connectedness Derivative Differential Geometry Differential Topology DIrect Limit Direct Limits Examples Exterior Algebra Field Theory Finite Dimensional Vector Spaces Full Solutions Functor Galois Theory Geometry Group Group Actions Group Algebra Groups Group Theory Halmos Homological Algebra Homomorphism Homotopy Ideals Induced Character Induced Representation Intuition Inverse Limit Irreducible Characters Irreps Isomorphism Linear Algebra Linear Transformations Manifolds Matrices Module Modules Module Theory Motivation Multivariable Analysis Munkres Normal Subgroups Number Theory Permutations PID Polynomials Prime Ideals Products Product Topology Projections Representation Theory Review of Group Theory Riemann Surfaces Ring Rings Ring Theory Rudin Solutions Sylow Theorems Symmetric Group Tensor Product Topology Total Derivative
[…] we appeal to the common fact that and so where here is Euler’s totient […]
Pingback by University of Maryland College Park Qualifying Exams (Group Theory and Representation Theory) ( January 2003)) « Abstract Nonsense  May 1, 2011 
[…] The result has already been proven for and so it suffices to prove this for . To see this note that there is a natural morphism which […]
Pingback by The Unit and Automorphism Group of Z/nZ « Abstract Nonsense  September 10, 2011 
[…] order and none of these are normal. This part is a bit more involved (for example, see this post on it), but the punch line is that it will be the cyclic group […]
Pingback by A Very Basic Introduction to the Sylow Theorems. « the dying love grape  May 18, 2012 