Automorphisms and Characters of Finite Groups Brittany Bianco, - PowerPoint PPT Presentation

B ACKGROUND : N OTATION P RE -T HEOREM B ACKGROUND : T HEOREM R ESULTS R EFERENCES Automorphisms and Characters of Finite Groups Brittany Bianco, Leigh Foster Mentor: Mandi A. Schaeffer Fry Metropolitan State University of Denver Nebraska Conference for Undergraduate Women in Mathematics January 26, 2019

B ACKGROUND : N OTATION P RE -T HEOREM B ACKGROUND : T HEOREM R ESULTS R EFERENCES B IG I DEA Fixed Notation ◮ G = Sp 4 ( q ) where q is a power of an odd prime, p ◮ H = { diag ( a , a − 1 , b , b − 1 ) | a , b ∈ F ∗ q } a subgroup of G ◮ ϕ m p is a “field automorphism” of G ◮ σ is an automorphism of Q ( e 2 π i / | G | ) Theorem Assume every ϕ m p -invariant member of Irr ( H ) is also fixed by σ . Then every ϕ m p -invariant member of Irr ( q − 1 ) ′ ( G ) is also fixed by σ .

B ACKGROUND : N OTATION P RE -T HEOREM B ACKGROUND : T HEOREM R ESULTS R EFERENCES Fixed Notation • G = Sp 4 ( q ) where q is a power of an odd prime, p • H = { diag ( a , a − 1 , b , b − 1 ) | a , b ∈ F ∗ q } a subgroup of G • ϕ m p is a “field automorphism” of G • σ is an automorphism of Q ( e 2 π i / | G | ) Sp 4 ( q ) = { g is an invertible 4 × 4 matrix over F q | g T Jg = J }   0 1 0 0 − 1 0 0 0   where J =   0 0 0 1   0 0 − 1 0

B ACKGROUND : N OTATION P RE -T HEOREM B ACKGROUND : T HEOREM R ESULTS R EFERENCES Fixed Notation • G = Sp 4 ( q ) where q is a power of an odd prime, p • H = { diag ( a , a − 1 , b , b − 1 ) | a , b ∈ F ∗ q } a subgroup of G • ϕ m p is a “field automorphism” of G • σ is an automorphism of Q ( e 2 π i / | G | ) By definition, a group ( G , ⋆ ) has:

B ACKGROUND : N OTATION P RE -T HEOREM B ACKGROUND : T HEOREM R ESULTS R EFERENCES Fixed Notation • G = Sp 4 ( q ) where q is a power of an odd prime, p • H = { diag ( a , a − 1 , b , b − 1 ) | a , b ∈ F ∗ q } a subgroup of G • ϕ m p is a “field automorphism” of G • σ is an automorphism of Q ( e 2 π i / | G | ) By definition, a group ( G , ⋆ ) has: ◮ Associativity ∀ a , b , c ∈ G , ( a ⋆ b ) ⋆ c = a ⋆ ( b ⋆ c )

B ACKGROUND : N OTATION P RE -T HEOREM B ACKGROUND : T HEOREM R ESULTS R EFERENCES Fixed Notation • G = Sp 4 ( q ) where q is a power of an odd prime, p • H = { diag ( a , a − 1 , b , b − 1 ) | a , b ∈ F ∗ q } a subgroup of G • ϕ m p is a “field automorphism” of G • σ is an automorphism of Q ( e 2 π i / | G | ) By definition, a group ( G , ⋆ ) has: ◮ Associativity ∀ a , b , c ∈ G , ( a ⋆ b ) ⋆ c = a ⋆ ( b ⋆ c ) ◮ An identity element, e ∃ e ∈ G s.t. ∀ a ∈ G , a ⋆ e = e ⋆ a = a .

B ACKGROUND : N OTATION P RE -T HEOREM B ACKGROUND : T HEOREM R ESULTS R EFERENCES Fixed Notation • G = Sp 4 ( q ) where q is a power of an odd prime, p • H = { diag ( a , a − 1 , b , b − 1 ) | a , b ∈ F ∗ q } a subgroup of G • ϕ m p is a “field automorphism” of G • σ is an automorphism of Q ( e 2 π i / | G | ) By definition, a group ( G , ⋆ ) has: ◮ Associativity ∀ a , b , c ∈ G , ( a ⋆ b ) ⋆ c = a ⋆ ( b ⋆ c ) ◮ An identity element, e ∃ e ∈ G s.t. ∀ a ∈ G , a ⋆ e = e ⋆ a = a . ◮ An inverse for every group element ∀ a ∈ G , ∃ b ∈ G (or a − 1 ) s.t. a ⋆ b = b ⋆ a = e

B ACKGROUND : N OTATION P RE -T HEOREM B ACKGROUND : T HEOREM R ESULTS R EFERENCES Fixed Notation • G = Sp 4 ( q ) where q is a power of an odd prime, p • H = { diag ( a , a − 1 , b , b − 1 ) | a , b ∈ F ∗ q } a subgroup of G • ϕ m p is a “field automorphism” of G • σ is an automorphism of Q ( e 2 π i / | G | ) By definition, a group ( G , ⋆ ) has: ◮ Associativity ∀ a , b , c ∈ G , ( a ⋆ b ) ⋆ c = a ⋆ ( b ⋆ c ) ◮ An identity element, e ∃ e ∈ G s.t. ∀ a ∈ G , a ⋆ e = e ⋆ a = a . ◮ An inverse for every group element ∀ a ∈ G , ∃ b ∈ G (or a − 1 ) s.t. a ⋆ b = b ⋆ a = e under the binary operation ⋆

B ACKGROUND : N OTATION P RE -T HEOREM B ACKGROUND : T HEOREM R ESULTS R EFERENCES Fixed Notation • G = Sp 4 ( q ) where q is a power of an odd prime, p • H = { diag ( a , a − 1 , b , b − 1 ) | a , b ∈ F ∗ q } a subgroup of G • ϕ m p is a “field automorphism” of G • σ is an automorphism of Q ( e 2 π i / | G | ) By definition, a group ( G , ⋆ ) has: ◮ Associativity ∀ a , b , c ∈ G , ( a ⋆ b ) ⋆ c = a ⋆ ( b ⋆ c ) ◮ An identity element, e ∃ e ∈ G s.t. ∀ a ∈ G , a ⋆ e = e ⋆ a = a . ◮ An inverse for every group element ∀ a ∈ G , ∃ b ∈ G (or a − 1 ) s.t. a ⋆ b = b ⋆ a = e under the binary operation ⋆ Example: Z 12 under addition

B ACKGROUND : N OTATION P RE -T HEOREM B ACKGROUND : T HEOREM R ESULTS R EFERENCES Fixed Notation • G = Sp 4 ( q ) where q is a power of an odd prime, p • H = { diag ( a , a − 1 , b , b − 1 ) | a , b ∈ F ∗ q } a subgroup of G • ϕ m p is a “field automorphism” of G • σ is an automorphism of Q ( e 2 π i / | G | ) By definition, a group ( G , ⋆ ) has: ◮ Associativity ∀ a , b , c ∈ G , ( a ⋆ b ) ⋆ c = a ⋆ ( b ⋆ c ) ◮ An identity element, e ∃ e ∈ G s.t. ∀ a ∈ G , a ⋆ e = e ⋆ a = a . ◮ An inverse for every group element ∀ a ∈ G , ∃ b ∈ G (or a − 1 ) s.t. a ⋆ b = b ⋆ a = e under the binary operation ⋆ Example: Z 12 under addition Non-Example: Z 12 under multiplication

B ACKGROUND : N OTATION P RE -T HEOREM B ACKGROUND : T HEOREM R ESULTS R EFERENCES Fixed Notation • G = Sp 4 ( q ) where q is a power of an odd prime, p • H = { diag ( a , a − 1 , b , b − 1 ) | a , b ∈ F ∗ q } a subgroup of G • ϕ m p is a “field automorphism” of G • σ is an automorphism of Q ( e 2 π i / | G | ) So Sp 4 ( q ) is a group?

B ACKGROUND : N OTATION P RE -T HEOREM B ACKGROUND : T HEOREM R ESULTS R EFERENCES Fixed Notation • G = Sp 4 ( q ) where q is a power of an odd prime, p • H = { diag ( a , a − 1 , b , b − 1 ) | a , b ∈ F ∗ q } a subgroup of G • ϕ m p is a “field automorphism” of G • σ is an automorphism of Q ( e 2 π i / | G | ) So Sp 4 ( q ) is a group? Recall Sp 4 ( q ) = { g is an invertible 4 × 4 matrix over F q | g T Jg = J } .

B ACKGROUND : N OTATION P RE -T HEOREM B ACKGROUND : T HEOREM R ESULTS R EFERENCES Fixed Notation • G = Sp 4 ( q ) where q is a power of an odd prime, p • H = { diag ( a , a − 1 , b , b − 1 ) | a , b ∈ F ∗ q } a subgroup of G • ϕ m p is a “field automorphism” of G • σ is an automorphism of Q ( e 2 π i / | G | ) So Sp 4 ( q ) is a group? Recall Sp 4 ( q ) = { g is an invertible 4 × 4 matrix over F q | g T Jg = J } . ◮ Associativity Matrix multiplication is associative

B ACKGROUND : N OTATION P RE -T HEOREM B ACKGROUND : T HEOREM R ESULTS R EFERENCES Fixed Notation • G = Sp 4 ( q ) where q is a power of an odd prime, p • H = { diag ( a , a − 1 , b , b − 1 ) | a , b ∈ F ∗ q } a subgroup of G • ϕ m p is a “field automorphism” of G • σ is an automorphism of Q ( e 2 π i / | G | ) So Sp 4 ( q ) is a group? Recall Sp 4 ( q ) = { g is an invertible 4 × 4 matrix over F q | g T Jg = J } . ◮ Associativity Matrix multiplication is associative ◮ An identity element, e e = I , the identity matrix since I T JI = J

B ACKGROUND : N OTATION P RE -T HEOREM B ACKGROUND : T HEOREM R ESULTS R EFERENCES Fixed Notation • G = Sp 4 ( q ) where q is a power of an odd prime, p • H = { diag ( a , a − 1 , b , b − 1 ) | a , b ∈ F ∗ q } a subgroup of G • ϕ m p is a “field automorphism” of G • σ is an automorphism of Q ( e 2 π i / | G | ) So Sp 4 ( q ) is a group? Recall Sp 4 ( q ) = { g is an invertible 4 × 4 matrix over F q | g T Jg = J } . ◮ Associativity Matrix multiplication is associative ◮ An identity element, e e = I , the identity matrix since I T JI = J ◮ An inverse for every group element Since g − 1 also satisfies the group definition: ( g − 1 ) T J ( g − 1 ) = J then every element has an inverse.

B ACKGROUND : N OTATION P RE -T HEOREM B ACKGROUND : T HEOREM R ESULTS R EFERENCES Fixed Notation • G = Sp 4 ( q ) where q is a power of an odd prime, p • H = { diag ( a , a − 1 , b , b − 1 ) | a , b ∈ F ∗ q } a subgroup of G • ϕ m p is a “field automorphism” of G • σ is an automorphism of Q ( e 2 π i / | G | ) A subgroup H is a subset of group elements of a group G that is itself a group under the group operation.

B ACKGROUND : N OTATION P RE -T HEOREM B ACKGROUND : T HEOREM R ESULTS R EFERENCES Fixed Notation • G = Sp 4 ( q ) where q is a power of an odd prime, p • H = { diag ( a , a − 1 , b , b − 1 ) | a , b ∈ F ∗ q } a subgroup of G • ϕ m p is a “field automorphism” of G • σ is an automorphism of Q ( e 2 π i / | G | ) A subgroup H is a subset of group elements of a group G that is itself a group under the group operation. Example: The evens mod 12 forms a subgroup of Z 12 under addition.

B ACKGROUND : N OTATION P RE -T HEOREM B ACKGROUND : T HEOREM R ESULTS R EFERENCES Fixed Notation • G = Sp 4 ( q ) where q is a power of an odd prime, p • H = { diag ( a , a − 1 , b , b − 1 ) | a , b ∈ F ∗ q } a subgroup of G • ϕ m p is a “field automorphism” of G • σ is an automorphism of Q ( e 2 π i / | G | ) A subgroup H is a subset of group elements of a group G that is itself a group under the group operation. Example: The evens mod 12 forms a subgroup of Z 12 under addition. Non-Example: The odds mod 12 do not form a subgroup of Z 12 under addition.