Bidirected edge-maximality of power graphs of finite cyclic groups - PowerPoint PPT Presentation

Bidirected edge-maximality of power graphs of finite cyclic groups Brian Curtin 1 Gholam Reza Pourgholi 2 1 Department of Mathematics and Statistics University of South Florida 2 School of Mathematics, Statistics and Computer Science University of Tehran Modern Trends in Algebraic Graph Theory Villanova University June 5th, 2014

Power Graphs G : finite group of order n

Power Graphs G : finite group of order n − → P ( G ) : directed power graph V ( G ) = G − → E ( G ) = { ( g , h ) | g , h ∈ G , h ∈ � g � − { g }}

Power Graphs G : finite group of order n − → P ( G ) : directed power graph V ( G ) = G − → E ( G ) = { ( g , h ) | g , h ∈ G , h ∈ � g � − { g }} bidirected edges ← → E ( G ) = {{ g , h } | ( g , h ) ∈ − → E ( G ) and ( h , g ) ∈ − → E ( G ) }

Power Graphs G : finite group of order n − → P ( G ) : directed power graph V ( G ) = G − → E ( G ) = { ( g , h ) | g , h ∈ G , h ∈ � g � − { g }} bidirected edges ← → E ( G ) = {{ g , h } | ( g , h ) ∈ − → E ( G ) and ( h , g ) ∈ − → E ( G ) } Lemma ∃ bidirected edge { g , h } iff g � = h generate the same subgroup.

Power Graphs − → P ( Z 6 ) 0

Power Graphs − → P ( Z 6 ) 3 0

Power Graphs − → P ( Z 6 ) 3 0

Power Graphs − → P ( Z 6 ) 2 3 0

Power Graphs − → P ( Z 6 ) 4 2 3 0

Power Graphs − → P ( Z 6 ) 4 2 3 0

Power Graphs − → P ( Z 6 ) 4 2 3 0

Power Graphs − → P ( Z 6 ) 4 2 3 0

Power Graphs − → P ( Z 6 ) 5 1 4 2 3 0

Power Graphs − → P ( Z 6 ) 5 1 4 2 3 0

Power Graphs − → P ( Z 6 ) 5 1 4 2 3 0

Power Graphs for groups of order 8 − → P (C 8 ) 1 2 3 0 4 5 6 7

Power Graphs for groups of order 8 → − − → P (C 8 ) P (C 4 × C 2 ) 1 11 2 01 3 31 0 4 00 20 5 10 6 21 7 30

Power Graphs for groups of order 8 → − → − − → P (C 8 ) P (C 2 × C 2 × C 2 ) P (C 4 × C 2 ) 1 11 110 2 01 001 3 31 101 0 4 00 20 000 010 5 10 011 6 21 100 7 30 111

Power Graphs for groups of order 8 → − → − − → P (C 8 ) P (C 2 × C 2 × C 2 ) P (C 4 × C 2 ) 1 11 110 2 01 001 3 31 101 0 4 00 20 000 010 5 10 011 6 21 100 7 30 111 − → P ( Q ) j i -j 1 -1 k -i -k

Power Graphs for groups of order 8 → − − → − → P (C 8 ) P (C 2 × C 2 × C 2 ) P (C 4 × C 2 ) 1 11 110 2 01 001 3 31 101 0 4 00 20 000 010 5 10 011 6 21 100 7 30 111 → − − → P ( D 8 ) P ( Q ) j ϕρ ϕρ 2 i ϕρ 3 -j 1 -1 e ϕ k ρ ρ 2 -i ρ 3 -k

Bidirectional edges Notation o ( g ) : order of g φ : Euler totient

Bidirectional edges Notation o ( g ) : order of g φ : Euler totient Count bidirected edges

Bidirectional edges Notation o ( g ) : order of g φ : Euler totient Count bidirected edges � g � : cyclic, order o ( g ),

Bidirectional edges Notation o ( g ) : order of g φ : Euler totient Count bidirected edges � g � : cyclic, order o ( g ), φ ( o ( g )) distinct generators

Bidirectional edges Notation o ( g ) : order of g φ : Euler totient Count bidirected edges � g � : cyclic, order o ( g ), φ ( o ( g )) distinct generators g is in φ ( o ( g )) − 1 bidirected edges

Bidirectional edges Notation o ( g ) : order of g φ : Euler totient Count bidirected edges � g � : cyclic, order o ( g ), φ ( o ( g )) distinct generators g is in φ ( o ( g )) − 1 bidirected edges Summing over G double counts

Bidirectional edges Notation o ( g ) : order of g φ : Euler totient Count bidirected edges � g � : cyclic, order o ( g ), φ ( o ( g )) distinct generators g is in φ ( o ( g )) − 1 bidirected edges Summing over G double counts Lemma |← → E ( G ) | = 1 � ( φ ( o ( g )) − 1) (1) 2 g ∈ G

A group sum Definition � φ ( G ) = φ ( o ( g )) . (2) g ∈ G

A group sum Definition � φ ( G ) = φ ( o ( g )) . (2) g ∈ G Corollary |← → E ( G ) | = φ ( G ) − | G | (3) 2

A group sum Definition � φ ( G ) = φ ( o ( g )) . (2) g ∈ G Corollary |← → E ( G ) | = φ ( G ) − | G | (3) 2 Notation C n : cyclic group of order n Compare φ ( G ), φ ( C n )

Results Main Theorem (BC, GR Pourgholi) Among finite groups of given order, the cyclic group has the maximum number of bidirectional edges in its directed power graph.

Results Main Theorem (BC, GR Pourgholi) Among finite groups of given order, the cyclic group has the maximum number of bidirectional edges in its directed power graph. Theorem (H Amiri, SM Jafarian Amiri, IM Isaacs) (Lindsey) Among finite groups of given order, the cyclic group has the maximum number of edges in its directed power graph.

Results Main Theorem (BC, GR Pourgholi) Among finite groups of given order, the cyclic group has the maximum number of bidirectional edges in its directed power graph. Theorem (H Amiri, SM Jafarian Amiri, IM Isaacs) (Lindsey) Among finite groups of given order, the cyclic group has the maximum number of edges in its directed power graph. Theorem (BC, GR Pourgholi) Among finite groups of given order, the cyclic group has the maximum number of edges in its undirected power graph.

Results restated Main Theorem (BC, GR Pourgholi) φ ( G ) ≤ φ ( C n )

Results restated Main Theorem (BC, GR Pourgholi) φ ( G ) ≤ φ ( C n ) i.e. � g ∈ G φ ( o ( g )) ≤ � z ∈ C n φ ( o ( z ))

Results restated Main Theorem (BC, GR Pourgholi) φ ( G ) ≤ φ ( C n ) i.e. � g ∈ G φ ( o ( g )) ≤ � z ∈ C n φ ( o ( z )) equality iff G ∼ = C n

Results restated Main Theorem (BC, GR Pourgholi) φ ( G ) ≤ φ ( C n ) i.e. � g ∈ G φ ( o ( g )) ≤ � z ∈ C n φ ( o ( z )) equality iff G ∼ = C n Theorem (H Amiri, SM Jafarian Amiri, IM Isaacs) (Lindsey) � g ∈ G o ( g ) ≤ � z ∈ C n o ( z ) equality iff G ∼ = C n

Results restated Main Theorem (BC, GR Pourgholi) φ ( G ) ≤ φ ( C n ) i.e. � g ∈ G φ ( o ( g )) ≤ � z ∈ C n φ ( o ( z )) equality iff G ∼ = C n Theorem (H Amiri, SM Jafarian Amiri, IM Isaacs) (Lindsey) � g ∈ G o ( g ) ≤ � z ∈ C n o ( z ) equality iff G ∼ = C n Theorem (BC, GR Pourgholi) � g ∈ G 2 o ( g ) − φ ( o ( g )) ≤ � z ∈ C n 2 o ( z ) − φ ( o ( z )), equality iff G ∼ = C n

φ ( C n ) Notation n = p α 1 1 p α 2 2 · · · p α k k p 1 < p 2 < · · · < p k primes α 1 , α 2 , . . . , α k ∈ Z +

φ ( C n ) Notation n = p α 1 1 p α 2 2 · · · p α k k p 1 < p 2 < · · · < p k primes α 1 , α 2 , . . . , α k ∈ Z + Lemma 2 α h d | n φ ( d ) 2 = � k p ( p h − 1)+2 φ (C n ) = � h h =1 p h +1

φ ( C n ) Notation n = p α 1 1 p α 2 2 · · · p α k k p 1 < p 2 < · · · < p k primes α 1 , α 2 , . . . , α k ∈ Z + Lemma 2 α h d | n φ ( d ) 2 = � k p ( p h − 1)+2 φ (C n ) = � h h =1 p h +1 Definition Q = � k p h +1 h =1 p h − 1

φ ( C n ) Notation n = p α 1 1 p α 2 2 · · · p α k k p 1 < p 2 < · · · < p k primes α 1 , α 2 , . . . , α k ∈ Z + Lemma 2 α h d | n φ ( d ) 2 = � k p ( p h − 1)+2 φ (C n ) = � h h =1 p h +1 Definition Q = � k p h +1 h =1 p h − 1 Lemma φ (C n ) > n 2 Q

An inequality If G is a counter example to main theorem: average of φ ( o ( g )) over G :

An inequality If G is a counter example to main theorem: average of φ ( o ( g )) over G : φ ( G ) ≥ φ (C n ) > n n n Q

An inequality If G is a counter example to main theorem: average of φ ( o ( g )) over G : φ ( G ) ≥ φ (C n ) > n n n Q ∃ g ∈ G with n < Q φ ( o ( g ))

An inequality If G is a counter example to main theorem: average of φ ( o ( g )) over G : φ ( G ) ≥ φ (C n ) > n n n Q ∃ g ∈ G with n < Q φ ( o ( g )) Key Theorem (technical proof) p : largest prime divisor of n If ∃ g ∈ G \{ id } st n < Q φ ( o ( g )), (as occurs if counterexample) Then ∃ normal (unique) Sylow p -subgroup P of G . P ⊆ � g � , so P cyclic.

Structure Theorem (Schur-Zassenhaus) If K ⊳ G with ( | K | , | G : K | ) = 1, then

Structure Theorem (Schur-Zassenhaus) If K ⊳ G with ( | K | , | G : K | ) = 1, then G = K ⋊ ϕ H (semidirect product) for some H ⊆ G and some homomorphism ϕ : H → Aut ( K ).

Structure Theorem (Schur-Zassenhaus) If K ⊳ G with ( | K | , | G : K | ) = 1, then G = K ⋊ ϕ H (semidirect product) for some H ⊆ G and some homomorphism ϕ : H → Aut ( K ). Corollary If ∃ g ∈ G \{ id } st n < Q φ ( o ( g )): G = P ⋊ ϕ H (semidirect product) P cyclic sylow p -group H subgroup with | P | , | H | coprime.

Semidirect products Lemma K : finite abelian group H : finite group, ( | K | , | H | ) = 1

Semidirect products Lemma K : finite abelian group H : finite group, ( | K | , | H | ) = 1 ∀ k ∈ K , h ∈ H o K ⋊ ϕ H ( kh ) | o K × H ( kh ) (order in semi direct product divides order in direct product)