Lecture 4.6: Automorphisms Matthew Macauley Department of - PowerPoint PPT Presentation

Lecture 4.6: Automorphisms Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4120, Modern Algebra M. Macauley (Clemson) Lecture 4.6: Automorphisms Math 4120, Modern Algebra 1 / 8

Basic concepts Definition An automorphism is an isomorphism from a group to itself. The set of all automorphisms of G forms a group, called the automorphism group of G , and denoted Aut( G ). Remarks. An automorphism is determined by where it sends the generators. An automorphism φ must send generators to generators. In particular, if G is cyclic, then it determines a permutation of the set of (all possible) generators. Examples 1. There are two automorphisms of Z : the identity, and the mapping n �→ − n . Thus, Aut( Z ) ∼ = C 2 . 2. There is an automorphism φ : Z 5 → Z 5 for each choice of φ (1) ∈ { 1 , 2 , 3 , 4 } . Thus, Aut( Z 5 ) ∼ = C 4 or V 4 . (Which one?) 3. An automorphism φ of V 4 = � h , v � is determined by the image of h and v . There are 3 choices for φ ( h ), and then 2 choices for φ ( v ). Thus, | Aut( V 4 ) | = 6, so it is either C 6 ∼ = C 2 × C 3 , or S 3 . (Which one?) M. Macauley (Clemson) Lecture 4.6: Automorphisms Math 4120, Modern Algebra 2 / 8

Automorphism groups of Z n Definition The multiplicative group of integers modulo n , denoted Z ∗ n or U ( n ), is the group U ( n ) := { k ∈ Z n | gcd( n , k ) = 1 } where the binary operation is multiplication, modulo n . U (5) = { 1 , 2 , 3 , 4 } ∼ U (8) = { 1 , 3 , 5 , 7 } ∼ = C 4 = C 2 × C 2 U (6) = { 1 , 5 } ∼ = C 2 1 2 3 4 1 3 5 7 1 1 2 3 4 1 5 1 1 3 5 7 2 2 4 1 3 1 1 5 3 3 1 7 5 3 3 1 4 2 5 5 1 5 5 7 1 3 4 4 3 2 1 7 7 5 3 1 Proposition (homework) The automorphism group of Z n is Aut( Z n ) = { σ a | a ∈ U ( n ) } ∼ = U ( n ), where σ a : Z n − → Z n , σ a (1) = a . M. Macauley (Clemson) Lecture 4.6: Automorphisms Math 4120, Modern Algebra 3 / 8

Automorphisms of D 3 Let’s find all automorphisms of D 3 = � r , f � . We’ll see a very similar example to this when we study Galois theory. Clearly, every automorphism φ is completely determined by φ ( r ) and φ ( f ). Since automorphisms preserve order, if φ ∈ Aut( D 3 ), then φ ( r ) = r or r 2 φ ( f ) = f , rf , or r 2 f φ ( e ) = e , , . � �� � � �� � 2 choices 3 choices Thus, there are at most 2 · 3 = 6 automorphisms of D 3 . Let’s try to define two maps, (i) α : D 3 → D 3 fixing r , and (ii) β : D 3 → D 3 fixing f : � α ( r ) = r � β ( r ) = r 2 α ( f ) = rf β ( f ) = f I claim that: these both define automorphisms (check this!) these generate six different automorphisms, and thus � α, β � ∼ = Aut( D 3 ). To determine what group this is isomorphic to, find these six automorphisms, and make a group presentation and/or multiplication table. Is it abelian? M. Macauley (Clemson) Lecture 4.6: Automorphisms Math 4120, Modern Algebra 4 / 8

Automorphisms of D 3 An automorphism can be thought of as a re-wiring of the Cayley diagram. e e e e r 2 f r 2 f f f id β → r 2 r �− → r r �− f f f �− → f f �− → f r r 2 r r 2 r 2 f rf r 2 f rf r 2 r 2 r r rf rf e e e e f r 2 f f r 2 f α αβ → r 2 r �− → r f f r �− → r 2 f f �− → rf f �− r r 2 r r 2 r 2 f rf r 2 f rf r 2 r r 2 r rf rf e e e e f r 2 f f r 2 f α 2 β α 2 → r 2 �− → r f f �− r r → r 2 f f �− → rf f �− r r 2 r r 2 r 2 f r 2 f rf rf r 2 r r 2 r rf rf M. Macauley (Clemson) Lecture 4.6: Automorphisms Math 4120, Modern Algebra 5 / 8

Automorphisms of D 3 Here is the multiplication table and Cayley diagram of Aut( D 3 ) = � α, β � . id αβ α 2 β α α 2 β id α 2 β α α 2 β αβ id id α 2 β α α α 2 αβ β id α 2 α 2 α α 2 β β αβ id α 2 β α 2 α β β αβ id α 2 β α α 2 αβ αβ β id α 2 β α 2 β αβ β α 2 α id It is purely coincidence that Aut( D 3 ) ∼ = D 3 . For example, we’ve already seen that Aut( Z 5 ) ∼ = U (5) ∼ Aut( Z 6 ) ∼ = U (6) ∼ Aut( Z 8 ) ∼ = U (8) ∼ = C 4 , = C 2 , = C 2 × C 2 . M. Macauley (Clemson) Lecture 4.6: Automorphisms Math 4120, Modern Algebra 6 / 8

Automorphisms of V 4 = � h , v � The following permutations are both automorphisms: α : v and β : v h hv h hv e e h β h id �− → h �− → v h h v �− → v v �− → h hv �− → hv hv �− → hv v v hv hv e α h e αβ h h �− → v �− → h h v �− → hv v �− → hv hv �− → h hv �− → v v v hv hv e α 2 h α 2 β e h �− → hv h h �− → hv v �− → h v �− → v hv �− → v v hv �− → h v hv hv M. Macauley (Clemson) Lecture 4.6: Automorphisms Math 4120, Modern Algebra 7 / 8

Automorphisms of V 4 = � h , v � Here is the multiplication table and Cayley diagram of Aut( V 4 ) = � α, β � ∼ = S 3 ∼ = D 3 . id αβ α 2 β α α 2 id β α 2 β α α 2 β αβ id id α 2 β α α α 2 αβ β id α 2 β α 2 α 2 α β αβ id α 2 β β β αβ α 2 α id α 2 β αβ αβ β α α 2 id α 2 β α 2 β αβ β α 2 α id hv and h Recall that α and β can be thought of as the permutations h v v hv → Perm( G ) ∼ and so Aut( G ) ֒ = S n always holds. M. Macauley (Clemson) Lecture 4.6: Automorphisms Math 4120, Modern Algebra 8 / 8