Bidirected edge-maximality of
power graphs of finite cyclic groups
Brian Curtin1
Gholam Reza Pourgholi2
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 ∈ hg i − {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 ∈ hg i − {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 ∈ hg i − {g }}
bidirected edges
←
→
→
−
→
−
E (G ) = {{g , h} | (g , h) ∈ E (G ) and (h, g ) ∈ E (G )}
Lemma
∃ bidirected edge {g , h} iff g 6= h generate the same subgroup.
Power Graphs
→
−
P (Z6 )
0
Power Graphs
→
−
P (Z6 )
3
0
Power Graphs
→
−
P (Z6 )
3
0
Power Graphs
→
−
P (Z6 )
2
3
0
Power Graphs
→
−
P (Z6 )
4
2
3
0
Power Graphs
→
−
P (Z6 )
4
2
3
0
Power Graphs
→
−
P (Z6 )
4
2
3
0
Power Graphs
→
−
P (Z6 )
4
2
3
0
Power Graphs
→
−
P (Z6 )
5
1
4
2
3
0
Power Graphs
→
−
P (Z6 )
5
1
4
2
3
0
Power Graphs
→
−
P (Z6 )
5
1
4
2
3
0
Power Graphs for groups of order 8
→
−
P (C8 )
1
2
3
0
4
5
6
7
Power Graphs for groups of order 8
→
−
P (C8 )
→
−
P (C4 × C2 )
11
1
01
2
31
3
0
00
4
20
10
5
21
6
7
30
Power Graphs for groups of order 8
→
−
P (C8 )
11
1
110
01
2
001
31
3
0
→
−
P (C2 ×C2 ×C2 )
→
−
P (C4 × C2 )
00
4
20
000
21
7
010
10
5
6
101
011
100
30
111
Power Graphs for groups of order 8
→
−
P (C8 )
11
1
110
01
2
001
31
3
0
→
−
P (C2 ×C2 ×C2 )
→
−
P (C4 × C2 )
00
4
20
000
010
10
5
21
6
→
−
P (Q)
j
i
-j
-1
k
-i
-k
011
100
30
7
1
101
111
Power Graphs for groups of order 8
→
−
P (C8 )
11
1
110
01
2
001
31
3
0
→
−
P (C2 ×C2 ×C2 )
→
−
P (C4 × C2 )
00
4
101
20
000
010
10
5
011
21
6
100
30
7
111
→
−
P (D8 )
→
−
P (Q)
ϕρ
j
ϕρ2
i
ϕρ3
-j
1
ϕ
e
-1
ρ
k
ρ2
-i
-k
ρ
3
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
hg i: cyclic, order o(g ),
Bidirectional edges
Notation
o(g ) : order of g
φ : Euler totient
Count bidirected edges
hg i: cyclic, order o(g ),
φ(o(g )) distinct generators
Bidirectional edges
Notation
o(g ) : order of g
φ : Euler totient
Count bidirected edges
hg i: 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
hg i: 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
hg i: cyclic, order o(g ),
φ(o(g )) distinct generators
g is in φ(o(g )) − 1 bidirected edges
Summing over G double counts
Lemma
←
→
1X
| E (G )| =
(φ(o(g )) − 1)
2
g ∈G
(1)
A group sum
Definition
φ(G ) =
X
g ∈G
φ(o(g )).
(2)
A group sum
Definition
φ(G ) =
X
φ(o(g )).
(2)
g ∈G
Corollary
←
→
φ(G ) − |G |
| E (G )| =
2
(3)
A group sum
Definition
φ(G ) =
X
φ(o(g )).
(2)
g ∈G
Corollary
←
→
φ(G ) − |G |
| E (G )| =
2
Notation
Cn : cyclic group of order n
Compare φ(G ), φ(Cn )
(3)
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(Cn )
Results restated
Main Theorem (BC, GR Pourgholi)
i.e.
φ(G ) ≤ φ(Cn )
P
g ∈G φ(o(g )) ≤
z∈Cn φ(o(z))
P
Results restated
Main Theorem (BC, GR Pourgholi)
φ(G ) ≤ φ(Cn )
P
i.e.
g ∈G φ(o(g )) ≤
z∈Cn φ(o(z))
equality iff G ∼
= Cn
P
Results restated
Main Theorem (BC, GR Pourgholi)
φ(G ) ≤ φ(Cn )
P
i.e.
g ∈G φ(o(g )) ≤
z∈Cn φ(o(z))
equality iff G ∼
= Cn
P
Theorem (H Amiri, SM Jafarian Amiri, IM Isaacs) (Lindsey)
P
g ∈G
equality iff G ∼
= Cn
o(g ) ≤
P
z∈Cn
o(z)
Results restated
Main Theorem (BC, GR Pourgholi)
φ(G ) ≤ φ(Cn )
P
i.e.
g ∈G φ(o(g )) ≤
z∈Cn φ(o(z))
equality iff G ∼
= Cn
P
Theorem (H Amiri, SM Jafarian Amiri, IM Isaacs) (Lindsey)
P
g ∈G
o(g ) ≤
P
z∈Cn
o(z)
z∈Cn
2o(z) − φ(o(z)),
equality iff G ∼
= Cn
Theorem (BC, GR Pourgholi)
P
2o(g ) − φ(o(g )) ≤
equality iff G ∼
= Cn
g ∈G
P
φ(Cn )
Notation
n = p1α1 p2α2 · · · pkαk
p1 < p2 < · · · < pk primes
α1 , α2 , . . . , αk ∈ Z+
φ(Cn )
Notation
n = p1α1 p2α2 · · · pkαk
p1 < p2 < · · · < pk primes
α1 , α2 , . . . , αk ∈ Z+
Lemma
φ(Cn ) =
2
d|n φ(d) =
P
Qk
h=1
2αh
ph
(ph −1)+2
ph +1
φ(Cn )
Notation
n = p1α1 p2α2 · · · pkαk
p1 < p2 < · · · < pk primes
α1 , α2 , . . . , αk ∈ Z+
Lemma
φ(Cn ) =
2
d|n φ(d) =
P
Definition
Q=
Qk
ph +1
h=1 ph −1
Qk
h=1
2αh
ph
(ph −1)+2
ph +1
φ(Cn )
Notation
n = p1α1 p2α2 · · · pkαk
p1 < p2 < · · · < pk primes
α1 , α2 , . . . , αk ∈ Z+
Lemma
φ(Cn ) =
2
d|n φ(d) =
P
Definition
Q=
Qk
ph +1
h=1 ph −1
Lemma
φ(Cn ) >
n2
Q
Qk
h=1
2αh
ph
(ph −1)+2
ph +1
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(Cn )
> Qn
n ≥
n
An inequality
If G is a counter example to main theorem:
average of φ(o(g )) over G :
φ(G )
φ(Cn )
> Qn
n ≥
n
∃ g ∈ G with n < Qφ(o(g ))
An inequality
If G is a counter example to main theorem:
average of φ(o(g )) over G :
φ(G )
φ(Cn )
> Qn
n ≥
n
∃ g ∈ G with n < Qφ(o(g ))
Key Theorem (technical proof)
p : largest prime divisor of n
If ∃ g ∈ G \{id} st n < Qφ(o(g )), (as occurs if counterexample)
Then ∃ normal (unique) Sylow p-subgroup P of G .
P ⊆ hg i, so P cyclic.
Structure
Theorem (Schur-Zassenhaus)
If K C G with (|K |, |G : K |) = 1, then
Structure
Theorem (Schur-Zassenhaus)
If K C G with (|K |, |G : K |) = 1, then
G = K oϕ H (semidirect product)
for some H ⊆ G and some homomorphism ϕ : H → Aut(K ).
Structure
Theorem (Schur-Zassenhaus)
If K C G with (|K |, |G : K |) = 1, then
G = K oϕ H (semidirect product)
for some H ⊆ G and some homomorphism ϕ : H → Aut(K ).
Corollary
If ∃ g ∈ G \{id} st n < Qφ(o(g )):
G = P oϕ 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
oK oϕ H (kh) | oK ×H (kh)
(order in semi direct product divides order in direct product)
Semidirect products
Lemma
K : finite abelian group
H : finite group, (|K |, |H|) = 1
∀ k ∈ K, h ∈ H
oK oϕ H (kh) | oK ×H (kh)
(order in semi direct product divides order in direct product)
Corollary
φ(K oϕ H) ≤ φ(K × H).
Semidirect products
Lemma
K : finite abelian group
H : finite group, (|K |, |H|) = 1
∀ k ∈ K, h ∈ H
oK oϕ H (kh) | oK ×H (kh)
(order in semi direct product divides order in direct product)
Corollary
φ(K oϕ H) ≤ φ(K × H).
Lemma
φ(K × H) ≤ φ(K )φ(H)
Semidirect products
Lemma
K : finite abelian group
H : finite group, (|K |, |H|) = 1
∀ k ∈ K, h ∈ H
oK oϕ H (kh) | oK ×H (kh)
(order in semi direct product divides order in direct product)
Corollary
φ(K oϕ H) ≤ φ(K × H).
Lemma
φ(K × H) ≤ φ(K )φ(H)
equality when (|K |, |H|) = 1.
Special case
Lemma. a, b : coprime positive integers
J := φ(Ca oϕ Cb ) ≤ φ(Ca × Cb ) =: H
equality iff the semi-direct product is direct.
Special case
Lemma. a, b : coprime positive integers
J := φ(Ca oϕ Cb ) ≤ φ(Ca × Cb ) =: H
equality iff the semi-direct product is direct.
Suppose equality holds. Then φ(oJ (g )) = φ(oH (g )) ∀g .
Special case
Lemma. a, b : coprime positive integers
J := φ(Ca oϕ Cb ) ≤ φ(Ca × Cb ) =: H
equality iff the semi-direct product is direct.
Suppose equality holds. Then φ(oJ (g )) = φ(oH (g )) ∀g .
Now oJ (g ) = oH (g ) or oJ (g ) = 2oH (g ) with oH (g ) odd.
Special case
Lemma. a, b : coprime positive integers
J := φ(Ca oϕ Cb ) ≤ φ(Ca × Cb ) =: H
equality iff the semi-direct product is direct.
Suppose equality holds. Then φ(oJ (g )) = φ(oH (g )) ∀g .
Now oJ (g ) = oH (g ) or oJ (g ) = 2oH (g ) with oH (g ) odd.
Suppose h generates H but not J.
Special case
Lemma. a, b : coprime positive integers
J := φ(Ca oϕ Cb ) ≤ φ(Ca × Cb ) =: H
equality iff the semi-direct product is direct.
Suppose equality holds. Then φ(oJ (g )) = φ(oH (g )) ∀g .
Now oJ (g ) = oH (g ) or oJ (g ) = 2oH (g ) with oH (g ) odd.
Suppose h generates H but not J. So m = oJ (h) = n/2 is odd.
Special case
Lemma. a, b : coprime positive integers
J := φ(Ca oϕ Cb ) ≤ φ(Ca × Cb ) =: H
equality iff the semi-direct product is direct.
Suppose equality holds. Then φ(oJ (g )) = φ(oH (g )) ∀g .
Now oJ (g ) = oH (g ) or oJ (g ) = 2oH (g ) with oH (g ) odd.
Suppose h generates H but not J. So m = oJ (h) = n/2 is odd.
Now L = hhi C J, |L| = m odd, |J : L| = 2.
Special case
Lemma. a, b : coprime positive integers
J := φ(Ca oϕ Cb ) ≤ φ(Ca × Cb ) =: H
equality iff the semi-direct product is direct.
Suppose equality holds. Then φ(oJ (g )) = φ(oH (g )) ∀g .
Now oJ (g ) = oH (g ) or oJ (g ) = 2oH (g ) with oH (g ) odd.
Suppose h generates H but not J. So m = oJ (h) = n/2 is odd.
Now L = hhi C J, |L| = m odd, |J : L| = 2.
Let K be a Sylow 2-subgroup of G , so |K | = 2.
Special case
Lemma. a, b : coprime positive integers
J := φ(Ca oϕ Cb ) ≤ φ(Ca × Cb ) =: H
equality iff the semi-direct product is direct.
Suppose equality holds. Then φ(oJ (g )) = φ(oH (g )) ∀g .
Now oJ (g ) = oH (g ) or oJ (g ) = 2oH (g ) with oH (g ) odd.
Suppose h generates H but not J. So m = oJ (h) = n/2 is odd.
Now L = hhi C J, |L| = m odd, |J : L| = 2.
Let K be a Sylow 2-subgroup of G , so |K | = 2.
Now J = LK and L ∩ K = {id}, so J = L oψ K ∼
= Cm oψ C2 .
Special case
Lemma. a, b : coprime positive integers
J := φ(Ca oϕ Cb ) ≤ φ(Ca × Cb ) =: H
equality iff the semi-direct product is direct.
Suppose equality holds. Then φ(oJ (g )) = φ(oH (g )) ∀g .
Now oJ (g ) = oH (g ) or oJ (g ) = 2oH (g ) with oH (g ) odd.
Suppose h generates H but not J. So m = oJ (h) = n/2 is odd.
Now L = hhi C J, |L| = m odd, |J : L| = 2.
Let K be a Sylow 2-subgroup of G , so |K | = 2.
Now J = LK and L ∩ K = {id}, so J = L oψ K ∼
= Cm oψ C2 .
Since J not cylic, it is dihedral group D2m .
Special case
Lemma. a, b : coprime positive integers
J := φ(Ca oϕ Cb ) ≤ φ(Ca × Cb ) =: H
equality iff the semi-direct product is direct.
Suppose equality holds. Then φ(oJ (g )) = φ(oH (g )) ∀g .
Now oJ (g ) = oH (g ) or oJ (g ) = 2oH (g ) with oH (g ) odd.
Suppose h generates H but not J. So m = oJ (h) = n/2 is odd.
Now L = hhi C J, |L| = m odd, |J : L| = 2.
Let K be a Sylow 2-subgroup of G , so |K | = 2.
Now J = LK and L ∩ K = {id}, so J = L oψ K ∼
= Cm oψ C2 .
Since J not cylic, it is dihedral group D2m .
Cm o 0 ∼
= Cm × 0 – same cogenerators of subgroups.
Special case
Lemma. a, b : coprime positive integers
J := φ(Ca oϕ Cb ) ≤ φ(Ca × Cb ) =: H
equality iff the semi-direct product is direct.
Suppose equality holds. Then φ(oJ (g )) = φ(oH (g )) ∀g .
Now oJ (g ) = oH (g ) or oJ (g ) = 2oH (g ) with oH (g ) odd.
Suppose h generates H but not J. So m = oJ (h) = n/2 is odd.
Now L = hhi C J, |L| = m odd, |J : L| = 2.
Let K be a Sylow 2-subgroup of G , so |K | = 2.
Now J = LK and L ∩ K = {id}, so J = L oψ K ∼
= Cm oψ C2 .
Since J not cylic, it is dihedral group D2m .
Cm o 0 ∼
= Cm × 0 – same cogenerators of subgroups.
Cm o 0, Cm × 0 can’t cogenerates w/ Cm o 1, Cm × 1, resp.
Special case
Lemma. a, b : coprime positive integers
J := φ(Ca oϕ Cb ) ≤ φ(Ca × Cb ) =: H
equality iff the semi-direct product is direct.
Suppose equality holds. Then φ(oJ (g )) = φ(oH (g )) ∀g .
Now oJ (g ) = oH (g ) or oJ (g ) = 2oH (g ) with oH (g ) odd.
Suppose h generates H but not J. So m = oJ (h) = n/2 is odd.
Now L = hhi C J, |L| = m odd, |J : L| = 2.
Let K be a Sylow 2-subgroup of G , so |K | = 2.
Now J = LK and L ∩ K = {id}, so J = L oψ K ∼
= Cm oψ C2 .
Since J not cylic, it is dihedral group D2m .
Cm o 0 ∼
= Cm × 0 – same cogenerators of subgroups.
Cm o 0, Cm × 0 can’t cogenerates w/ Cm o 1, Cm × 1, resp.
Cm o 1 flips, no cogenerators, Cm × 1 has all generators of C2m
Special case
Lemma. a, b : coprime positive integers
J := φ(Ca oϕ Cb ) ≤ φ(Ca × Cb ) =: H
equality iff the semi-direct product is direct.
Suppose equality holds. Then φ(oJ (g )) = φ(oH (g )) ∀g .
Now oJ (g ) = oH (g ) or oJ (g ) = 2oH (g ) with oH (g ) odd.
Suppose h generates H but not J. So m = oJ (h) = n/2 is odd.
Now L = hhi C J, |L| = m odd, |J : L| = 2.
Let K be a Sylow 2-subgroup of G , so |K | = 2.
Now J = LK and L ∩ K = {id}, so J = L oψ K ∼
= Cm oψ C2 .
Since J not cylic, it is dihedral group D2m .
Cm o 0 ∼
= Cm × 0 – same cogenerators of subgroups.
Cm o 0, Cm × 0 can’t cogenerates w/ Cm o 1, Cm × 1, resp.
Cm o 1 flips, no cogenerators, Cm × 1 has all generators of C2m
equality fails, contradiction unless s.d.p. is direct.
Outline of proof of main result
Strategy
Induct on number of prime factors.
Outline of proof of main result
Strategy
Induct on number of prime factors.
Launch with prime powers–counterexample would be cyclic.
Outline of proof of main result
Strategy
Induct on number of prime factors.
Launch with prime powers–counterexample would be cyclic.
If G counterexample:
G = P oϕ H
Outline of proof of main result
Strategy
Induct on number of prime factors.
Launch with prime powers–counterexample would be cyclic.
If G counterexample:
G = P oϕ H
P cyclic sylow p-group, (|P|, |H|) = 1,
|H| fewer prime divors than |G |
Outline of proof of main result
Strategy
Induct on number of prime factors.
Launch with prime powers–counterexample would be cyclic.
If G counterexample:
G = P oϕ H
P cyclic sylow p-group, (|P|, |H|) = 1,
|H| fewer prime divors than |G |
φ(G ) ≤ φ(P × H) = φ(P)φ(H)
Outline of proof of main result
Strategy
Induct on number of prime factors.
Launch with prime powers–counterexample would be cyclic.
If G counterexample:
G = P oϕ H
P cyclic sylow p-group, (|P|, |H|) = 1,
|H| fewer prime divors than |G |
φ(G ) ≤ φ(P × H) = φ(P)φ(H)
φ(H) ≤ φ(C|H| ), equal iff H cyclic
Outline of proof of main result
Strategy
Induct on number of prime factors.
Launch with prime powers–counterexample would be cyclic.
If G counterexample:
G = P oϕ H
P cyclic sylow p-group, (|P|, |H|) = 1,
|H| fewer prime divors than |G |
φ(G ) ≤ φ(P × H) = φ(P)φ(H)
φ(H) ≤ φ(C|H| ), equal iff H cyclic
H must be cyclic
Outline of proof of main result
Strategy
Induct on number of prime factors.
Launch with prime powers–counterexample would be cyclic.
If G counterexample:
G = P oϕ H
P cyclic sylow p-group, (|P|, |H|) = 1,
|H| fewer prime divors than |G |
φ(G ) ≤ φ(P × H) = φ(P)φ(H)
φ(H) ≤ φ(C|H| ), equal iff H cyclic
H must be cyclic
G∼
= C|P| oϕ C|H| & Cn ∼
= C|P| × C|H|
Outline of proof of main result
Strategy
Induct on number of prime factors.
Launch with prime powers–counterexample would be cyclic.
If G counterexample:
G = P oϕ H
P cyclic sylow p-group, (|P|, |H|) = 1,
|H| fewer prime divors than |G |
φ(G ) ≤ φ(P × H) = φ(P)φ(H)
φ(H) ≤ φ(C|H| ), equal iff H cyclic
H must be cyclic
G∼
= C|P| oϕ C|H| & Cn ∼
= C|P| × C|H|
φ(C|P| oϕ φ(C|H| ) = φ(C|P| × C|H| ) iff s.d.p. is direct
Outline of proof of main result
Strategy
Induct on number of prime factors.
Launch with prime powers–counterexample would be cyclic.
If G counterexample:
G = P oϕ H
P cyclic sylow p-group, (|P|, |H|) = 1,
|H| fewer prime divors than |G |
φ(G ) ≤ φ(P × H) = φ(P)φ(H)
φ(H) ≤ φ(C|H| ), equal iff H cyclic
H must be cyclic
G∼
= C|P| oϕ C|H| & Cn ∼
= C|P| × C|H|
φ(C|P| oϕ φ(C|H| ) = φ(C|P| × C|H| ) iff s.d.p. is direct
contradiction; no counterexamples.
A group sum inequality and its application to power graphs
To appear
Bulletin of the Australian Mathematical Society
A group sum inequality and its application to power graphs
To appear
Bulletin of the Australian Mathematical Society
ArXiv
arXiv:1311.2983
A group sum inequality and its application to power graphs
To appear
Bulletin of the Australian Mathematical Society
ArXiv
arXiv:1311.2983
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