The Classification of Finite Groups of Order 16 Kyle Whitcomb Department of Mathematics and Computer Science University of Puget Sound Tacoma, Washington May 5, 2015
Finite Groups of Order 16 Outline 1 Definitions and Notation 2 Preliminary Theorems and Calculations 3 Restricting the Possible Extension Types The Big Theorem The Big (Abridged) Proof 4 The Finite Groups of Order 16
Finite Groups of Order 16 Introduction There are significantly more groups of order 16 than of groups with lesser order. To put it more precisely, here is a table with the number of groups with orders 2 to 16: Order n 1 2 3 4 5 6 7 8 # groups with order n 1 1 1 2 1 2 1 5 Order n 9 10 11 12 13 14 15 16 # groups with order n 2 2 1 5 1 2 1 14 We seek to classify all 14 of these groups of order 16 by utilizing extension types.
Finite Groups of Order 16 Definitions and Notation Table of Contents 1 Definitions and Notation 2 Preliminary Theorems and Calculations 3 Restricting the Possible Extension Types The Big Theorem The Big (Abridged) Proof 4 The Finite Groups of Order 16
Finite Groups of Order 16 Definitions and Notation Familiar Concepts Familiar Concepts We will rely on the previous knowledge of the following concepts in abstract algebra, which we should be familiar with from Judson’s Abstract Algebra . Abelian groups Normal subgroups, N ⊳ G Generators, and groups generated by multiple elements, G = � g 1 , g 2 , . . . � Centers, Z ( G ) Automorphisms, and the automorphism group Aut( G )
Finite Groups of Order 16 Definitions and Notation Familiar Concepts Familiar Concepts (cont.) The inner automorphism For a ∈ G , there is an inner automorphism of G , t a : G → G , t a ( x ) = axa − 1 Conjugate elements Two elements, g 1 , g 2 ∈ G , are conjugate if there exists an inner automorphism t a of G such that t a ( g 1 ) = g 2 .
Finite Groups of Order 16 Definitions and Notation Semidirect Products Inner Semidirect Products The inner semidirect product is a very easy construction if you recall the inner direct product. Definition Given a group G , if N ⊳ G and H ⊆ G such that 1 G = NH = { nh | n ∈ N , h ∈ H } , and 2 N ∩ H = { e G } , then G is the inner semidirect product of N and H .
Finite Groups of Order 16 Definitions and Notation Semidirect Products Outer Semidirect Products If G is an inner semidirect product of N and H , then G is isomorphic to an outer semidirect product of N and H , G ∼ = N ⋊ ϕ H .
Finite Groups of Order 16 Definitions and Notation Semidirect Products Outer Semidirect Products If G is an inner semidirect product of N and H , then G is isomorphic to an outer semidirect product of N and H , G ∼ = N ⋊ ϕ H . Definition N and H are groups, and ϕ is a homomorphism ϕ : H → Aut( N ), ϕ ( h ) = ϕ h where ϕ h ( n ) = hnh − 1 for h ∈ H , n ∈ N . The outer semidirect product of N and H with respect to ϕ is N ⋊ ϕ H , where the operation is ∗ : ( N × H ) × ( N × H ) → N ⋊ ϕ H , ( n 1 , h 1 ) ∗ ( n 2 , h 2 ) = ( n 1 ϕ h 1 ( n 2 ) , h 1 h 2 ) .
Finite Groups of Order 16 Definitions and Notation Cyclic Extensions and Extension Types Cyclic Extensions Definition (Cyclic Extension) Let N ⊳ G . If G / N ∼ = Z n , then G is a cyclic extension of N .
Finite Groups of Order 16 Definitions and Notation Cyclic Extensions and Extension Types Some Properties of Cyclic Extensions Suppose G is a cyclic extension of N , G / N ∼ = Z n .
Finite Groups of Order 16 Definitions and Notation Cyclic Extensions and Extension Types Some Properties of Cyclic Extensions Suppose G is a cyclic extension of N , G / N ∼ = Z n . Consider a ∈ G such that | Na | = n in G / N , then v = a n ∈ N .
Finite Groups of Order 16 Definitions and Notation Cyclic Extensions and Extension Types Some Properties of Cyclic Extensions Suppose G is a cyclic extension of N , G / N ∼ = Z n . Consider a ∈ G such that | Na | = n in G / N , then v = a n ∈ N . Consider τ ∈ Aut( N ) such that τ is the restriction to N of the inner automorphism t a of G .
Finite Groups of Order 16 Definitions and Notation Cyclic Extensions and Extension Types Some Properties of Cyclic Extensions Suppose G is a cyclic extension of N , G / N ∼ = Z n . Consider a ∈ G such that | Na | = n in G / N , then v = a n ∈ N . Consider τ ∈ Aut( N ) such that τ is the restriction to N of the inner automorphism t a of G . Then τ ( v ) = ava − 1 = aa n a − 1 = a 1+ n − 1 = a n = v and τ n ( x ) = aa · · · a ( x ) a − 1 · · · a − 1 a − 1 = a n xa − n = vxv − 1 = t v ( x ) for all x ∈ N . Therefore τ n = t v .
Finite Groups of Order 16 Definitions and Notation Cyclic Extensions and Extension Types Extension Types Definition For a group N , a quadruple ( N , n , τ, v ) is an extension type if v ∈ N , τ ∈ Aut( N ), τ ( v ) = v , and τ n = t v .
Finite Groups of Order 16 Definitions and Notation Cyclic Extensions and Extension Types Extension Types Definition For a group N , a quadruple ( N , n , τ, v ) is an extension type if v ∈ N , τ ∈ Aut( N ), τ ( v ) = v , and τ n = t v . Definition Given a group G , if 1 N ⊳ G , 2 G / N ∼ = Z n , 3 there exists a ∈ G such that v = a n , 4 and there exists τ ∈ Aut( G ) such that τ n = t v and τ ( v ) = v , then G realizes the extension type ( N , n , τ, v ).
Finite Groups of Order 16 Preliminary Theorems and Calculations Table of Contents 1 Definitions and Notation 2 Preliminary Theorems and Calculations 3 Restricting the Possible Extension Types The Big Theorem The Big (Abridged) Proof 4 The Finite Groups of Order 16
Finite Groups of Order 16 Preliminary Theorems and Calculations Equivalence of Extension Types Equivalence of Extension Types Theorem Two extension types, ( N , n , τ, v ) and ( N ′ , n , σ, w ) are equivalent if there exists an isomorphism ϕ : N → N ′ such that σ = ϕτϕ − 1 and w = ϕ ( v ) .
Finite Groups of Order 16 Preliminary Theorems and Calculations Isomorphic Groups Realize Equivalent Extension Types Isomorphic Groups Realize Equivalent Extension Types Theorem G realizes ( N , n , τ, v ) and H realizes ( M , n , σ, w ) . If ( N , n , τ, v ) ∼ ( M , n , σ, w ) , then G ∼ = H.
Finite Groups of Order 16 Preliminary Theorems and Calculations Subgroups of Groups of Order 16 Important Subgroups of Groups of Order 16 Outlier group: Z 4 2 Theorem If | G | = 16 and G ≇ Z 4 2 , then either Z 8 ⊳ G or K 8 ⊳ G, where K 8 ≡ Z 4 × Z 2 .
Finite Groups of Order 16 Preliminary Theorems and Calculations Automorphisms of Z 8 and K 8 Automorphisms of Z 8 If α is a generator of Z 8 , Z 8 = � α � , then all of the automorphisms of Z 8 can be expressed as follows. Automorphism φ i ∈ Aut( Z 8 ) φ i ( α ) φ 1 α α 3 φ 2 α 5 φ 3 α 7 φ 4
Finite Groups of Order 16 Preliminary Theorems and Calculations Automorphisms of Z 8 and K 8 Automorphisms of K 8 Similarly, if Z 4 = � β � and Z 2 = � γ � , then K 8 = � β, γ � . The automorphisms of K 8 are then: Automorphism ψ i ∈ Aut( K 8 ) ψ i ( β ) ψ i ( γ ) ψ 1 β γ β 3 γ β 2 γ ψ 2 β 3 ψ 3 γ β 2 γ ψ 4 βγ ψ 5 βγ γ β 3 β 2 γ ψ 6 β 3 γ ψ 7 γ β 2 γ ψ 8 β
Finite Groups of Order 16 Restricting the Possible Extension Types Table of Contents 1 Definitions and Notation 2 Preliminary Theorems and Calculations 3 Restricting the Possible Extension Types The Big Theorem The Big (Abridged) Proof 4 The Finite Groups of Order 16
Finite Groups of Order 16 Restricting the Possible Extension Types The Big Theorem The Big Theorem Theorem Every group G of order 16 that is not isomorphic to Z 4 2 realizes one of the following extension types, where Z 8 = � α � and K 8 = � β, γ � : ( Z 8 , 2 , φ 1 , e ) , ( Z 8 , 2 , φ 2 , e ) ( Z 8 , 2 , φ 3 , e ) , ( Z 8 , 2 , φ 4 , e ) , ( Z 8 , 2 , φ 4 , α 4 ) , ( Z 8 , 2 , φ 1 , α ) , ( K 8 , 2 , ψ 1 , e ) , ( K 8 , 2 , ψ 3 , e ) , ( K 8 , 2 , ψ 3 , β 2 ) , ( K 8 , 2 , ψ 5 , β 2 ) , ( K 8 , 2 , ψ 5 , e ) , ( K 8 , 2 , ψ 6 , e ) , ( K 8 , 2 , ψ 1 , γ ) .
Finite Groups of Order 16 Restricting the Possible Extension Types The Big (Abridged) Proof The Big Proof Proof Skeleton: Preliminary details Case 1. Case 2. { Subcases i, ii, iii } Case 3. Case 4. Case 5. Case 6. { Subcases i, ii, iii }
Finite Groups of Order 16 Restricting the Possible Extension Types The Big (Abridged) Proof Excerpts from The Big Proof Preliminary setup: For G ≇ Z 4 2 , K 8 ⊳ G or Z 8 ⊳ G
Finite Groups of Order 16 Restricting the Possible Extension Types The Big (Abridged) Proof Excerpts from The Big Proof Preliminary setup: For G ≇ Z 4 2 , K 8 ⊳ G or Z 8 ⊳ G [ G : Z 8 ] = [ G : K 8 ] = 2, so n = 2
Finite Groups of Order 16 Restricting the Possible Extension Types The Big (Abridged) Proof Excerpts from The Big Proof Preliminary setup: For G ≇ Z 4 2 , K 8 ⊳ G or Z 8 ⊳ G [ G : Z 8 ] = [ G : K 8 ] = 2, so n = 2 All possible extension types (up to isomorphism) take the form ( K 8 , 2 , ψ i , v ) and ( Z 8 , 2 , φ j , v )
Finite Groups of Order 16 Restricting the Possible Extension Types The Big (Abridged) Proof Excerpts from The Big Proof Preliminary setup: For G ≇ Z 4 2 , K 8 ⊳ G or Z 8 ⊳ G [ G : Z 8 ] = [ G : K 8 ] = 2, so n = 2 All possible extension types (up to isomorphism) take the form ( K 8 , 2 , ψ i , v ) and ( Z 8 , 2 , φ j , v ) v = g 2 for some inducing element g ∈ G
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