An approximation of the number of subgroups of
a finite group
Carlos Segovia González
Universidad de los Andes, Bogotá Colombia
Mayo 2012
1
2
3
An unsolve problem in geometric groups theory is to give an
explicit formula for the number of subgroups of a (finite)
non-Abelian group.
The number of subgroups of the cyclic group Zn is the
number of divisors of the integer n, denoted by τ (n).
For other abelian groups, this is a complicated task, see
Grigore Cǎlugǎreanu, The total number of subgroups of a
finite Abelian group, Scientiae Mathematicae Japonicae.
Marius Tǎrnǎuceanu, An arithmetic method of counting the
subgroups of a finite abelian group, Bull. Math. Soc. Sci.
Math. Roumanie.
G.A. Miller, Subgroups of the groups whose orders below
thirty, Proc. N.A.S.
Stephan Cavior, in The subgroups of the dihedral group
(Math. Mag.), proved that the number of subgroups is given
by τ (n) + σ(n), where σ(n) is the sum of the divisors.
“Commutator length is the algebraic analogue of “number of
handles” in group theory......Danny Calegari”
Definition (G -Frobenius algebra)
Definition (G -cobordism category)
A is an algebra A = ⊕g ∈G Ag , where Ag is a vector
space of finite dimension, for all g ∈ G , such that
A homomorphism α : G −→ Aut(A) .
2
Products mg ,h : Ag ⊗ Ah −→ Agh such that
for all x ∈ Ag and y ∈ Ah we have
The objects are based G -principal bundles over
the circle (up to diffeomorphisms.) boundary)
>
1
1
g
xy = αg (y )x .
The morphisms are G -cobordisms (up to
diffeomorphisms that restrict to the identity on
the boundary.)
e
g
g
e
g
g
g
g
g
g
g
= =
A G -invariant trace ϕ : Ae −→ C and
coassociative coproducts
∆g ,h : Agh −→ Ag ⊗ Ah , with ϕ as its
counit when g or h is the identity, such that
for every g ∈ G the composition
= =
2
3
kgk -1
g
g
g -1
g
k
g
g
ϕ ◦ ∆g ,g −1
is non-degenrate.
4
g
The Euler element
P g
g −1
∈ Ag ⊗ Ag −1 ,
∆g ,g −1 =
i ξi ⊗ ξi
n o
g
g −1
where ξi
is a base of Ag and ξi
is
h
g
e
g
e
h
e
e
e
hg
gh
kgk -1
g
g
i
g
g −1
αk (ξi )ξi
i
kgk -1
=
h = [k,g]
g -1
k
k
k
X k
k −1
=
ξi αg (ξi
).
hg
h
g
h
the dual base of Ag −1 . We have that for all
g , k ∈ G the following identity holds
X
hgh -1
h
=
g
g -1
k -1
gk -1g-1
The equations
MG = {(g1 , k1 )...(gm , km ) : [km , g1 ]...[k1 , g1 ] = 1}/ ∼
MG =
g1




1


g2
k1
gn
k2
kn
1
[k1,g1]
g1-1
[k2,g2][k1,g1]
g2-1[k1,g1]
/∼










1






The action of axiom 4
(g , k) ∼ (gkg
−1
,g
−1
) and (g , k) ∼ (k, g
−1
).
1
kgk -1
g
k
k
2
The action of axiom 1
=
g
g -1
m
(g , k) ∼ (g , kg ) .
k -1
gk -1g-1
2
e
g
(g1 , k)(g2 , k
3
−1
−1
) ∼ (g1
,k
−1
)((k
−1 2 −1 2
) g2 k , k) .
The action of axiom 3 induces the two
equations
0
0
0
0
0
(g , k)(g , k ) ∼ ([g , k]g , k )k (g , k)k
= =
g
0
−1
0
0
(g , k )k(k
−1
g
g
g
g
g
g
g -1
g
g
g
g
3
g
0−1
[k, g ]kg , k)
hk
h
k
g
g
g
hk
hk
gh
=
0
(g , k)(g , k ) ∼ k
e
g
= =
and for elements of the form (g1 , k)(g2 , k −1 ),
we get the following
gh
=
h
k
hk
1
G
ORDERS
DESCRIPTION OF THE GROUPS
4
6
8
Cyclic(Z4 ), Z22
3,5
Z6 , symmetric(Σ3 )
4,6
Z8 , octic(D8 ), quaternion(Q8 ) 4,10,6
Z4 × Z2 , Z32
8,16
Z9 , Z23
3,6
Z10 , dihedral(D10 )
4,8
Z12 , tetrahedral(A4 ), D12
6,10,16
Dicyclic(Dic3 ), Z22 × Z3
8,10
Z14 , D14
4,10
Z15
4
Z16 , Dic4 , D16 , (Q8 × Z2 )
5,11,19,19
2
2
4
Z8 × Z2 , Z4 , Z4 × Z2 , Z2
11,15,27,67
Modular group of order 16
11
Quasihedral of order 16
15
D8 o Z2
35
(Z4 × Z2 ) o Z2
23
G4,4
15
Q8 o Z2
23
Z18 , Z3 × Z6 , D18
6,12,16
(Z3 × Z3 ) o Z2 , Σ3 × Z3
28,14
Z20 , Z10 × Z2 , D20
6,10,22
Dic5 , metacyclic
10,14
Z21 , Z7 o Z3
4,10
Z22 , D22
4,14
Z24 ,Z2 × Z12 ,Z2 × Z2 × Z6
8,16,32
D8 × Z3 ,Q8 × Z3
20,12
Sl(2, 3), A4 × Z2
15,26
Σ4 , D24 , Dic6
30,34,18
Z2 × Z2 × Σ3 , Z2 × (Z3 o Z4 ) 54,22
Z4 × Σ3 ,Z3 o Z8
26,10
(Z6 × Z2 ) o Z2
30
Z25 , Z25
3,8
Z26 , D26
4,16
3
Z27 , Z9 × Z3 , Z3
4,10,28
(Z3 × Z3 ) o Z3 , Z9 o Z3
19,10
Z28 , Z14 × Z2 , D28 , Dic7
6,10,28,12
9
10
12
14
15
16
18
20
21
22
24
25
26
27
28
SUBGROUPS
ABE-SUBGROUPS
GENERATOR’S
3,5
4,5
4,9,5
8,16
3,6
4,7
6,9,13
7,10
4,9
4
5,8,16,14
11,15,27,67
10
12
30
22
14
18
6,12,12
15,12
6,10,19
9,12
4,9
4,13
8,16,32
18,10
13,24
21,24,12
43,19
21,9
22
3,8
4,15
4,10,28
18,9
6,10,25,11
3,5
4,5+1
4,9+1,5+1
8,15
2
3,7
4,7+1
6,9+1,13+14+
7+1,10
4,9+1
4
5,8+4+ ,16+4+ ,14+10+
11,16,25,51
10+1
12+4+
28+
21+
10+7+
20+
6,14,12+
3
16+ ,13+13
6,10 ,19+
9+1, 12+3+
4,9+1
4,13+4+
8,16,30
18+ ,10+
13+ ,23+
21+3+ ,24+ ,12+
40+ ,19+
21+ ,9+
22+
3,11
4,15+
4,12,40
22+ ,10+
4
6,10,25+ ,11+
5
MG ∼
=
r
M
N
i=1
The classifying space of the G -cobordism
category has the homotopy type
G
[G , G ]
× XG × T
rG
×G EG ,
where X G is the homotopy fiber of a suitable
map Φ, and T r
G
×G EG is the Borel
construction with T r
G
the r G -torus.
The connected components and the homotopy
groups are given by
G
π0 (S ) ∼
=
G
[G , G ]
r
G
π1 (S ) ∼
= Z
G
,
oG.
G
πn (S ) ∼
= πn (XG ) .
With the use of MATLAB we obtain this table.
With the use of The On-Line Encyclopedia of
Integer Sequences (OEIS), we obtain some
consequences.
ORDERS
4
6
8
9
10
12
14
15
16
18
20
21
22
24
25
26
27
28
DESCRIPTION OF THE GROUPS
Cyclic(Z4 ) , Z22
SUBGROUPS
ABE-SUBGROUPS
3,5
3,5
Z6 , symmetric(Σ3 )
4,6
Z8 , octic(D8 ), quaternion(Q84,10,6
)
Z4 × Z2 , Z32
8,16
Z9 , Z23
3,6
Z10 , dihedral(D10 )
4,8
Z12 , tetrahedral(A4 ), D12
6,10,16
Dicyclic(Dic3 ), Z22 × Z3
8,10
Z14 , D14
4,10
Z15
4
Z16 , Dic4 , D16 , (Q8 × Z2 ) 5,11,19,19
2
2
4
Z8 × Z2 , Z4 , Z4 × Z2 , Z2
11,15,27,67
Modular group of order 16
11
Quasihedral of order 16
15
D8 o Z2
35
(Z4 × Z2 ) o Z2
23
G4,4
15
Q8 o Z2
23
Z18 , Z3 × Z6 , D18
6,12,16
(Z3 × Z3 ) o Z2 , Σ3 × Z3
28,14
Z20 , Z10 × Z2 , D20
6,10,22
Dic5 , metacyclic
10,14
Z21 , Z7 o Z3
4,10
Z22 , D22
4,14
Z24 ,Z2 × Z12 ,Z2 × Z2 × Z6 8,16,32
D8 × Z3 ,Q8 × Z3
20,12
Sl(2, 3), A4 × Z2
15,26
Σ4 , D24 , Dic6
30,34,18
Z2 × Z2 × Σ3 , Z2 × (Z3 o Z4 ) 54,22
Z4 × Σ3 ,Z3 o Z8
26,10
(Z6 × Z2 ) o Z2
30
Z25 , Z25
3,8
Z26 , D26
4,16
Z27 , Z9 × Z3 , Z33
4,10,28
(Z3 × Z3 ) o Z3 , Z9 o Z3
19,10
Z28 , Z14 × Z2 , D28 , Dic7
6,10,28,12
4,5
4,9,5
8,16
3,6
4,7
6,9,13
7,10
4,9
4
5,8,16,14
11,15,27,67
10
12
30
22
14
18
6,12,12
15,12
6,10,19
9,12
4,9
4,13
8,16,32
18,10
13,24
21,24,12
43,19
21,9
22
3,8
4,15
4,10,28
18,9
6,10,25,11
GENERATOR’S
3 ,5
1
4 ,5+1
4 ,9+1,5+1
8,15
3 ,7
4,7+1
6 ,9+1,13+14+
7+1,10
4 ,9+1
4
5 ,8+4+ ,16+4+ ,14+10+
11,16,25,51
2
10+1
12+4+
28+
21+
10+7+
20+
6 ,14,12+
16+ ,13+13
6 ,10 ,19+
9+1, 12+3+
4 ,9+1
4 ,13+4+
8 ,16,30
18+ ,10+
13+ ,23+
21+3+ ,24+ ,12+
40+ ,19+
21+ ,9+
22+
3 ,11
4 ,15+
4 ,12,40
22+ ,10+
6 ,10,25+ ,11+
The equations reduce as follows:
(g , k) ∼ (k, −g )
(g1 , k1 )(g2 , k2 ) ∼ (g2 , k2 )(g1 , k1 )
(g , k) ∼ (g , mg + k)
where m ∈ Z.
For the cyclic groups Zn , the number r Zn
coincides with the number of subgroups of Zn .
This is just a consequence of division
algorithm.
k = q1 g + r1
(1)
−g = q2 r1 + r2
(2)
r1 = q3 r2 + r3
(3)
.
.
.
±rm−1 = qm+1 rm
with 0 ≤ |rm−1 | < ... < |r1 | < |k|.
(g , k) ∼ (g , r1 ) ∼ (r1 , −g ) ∼ (r1 , r2 ) ∼
∼ (r2 , −r1 ) ∼ (r2 , −r3 ) ∼ · · · ∼
(rm−1 , rm−2 ) ∼ (rm1 , −qrm−1 ) ∼
(rm−1 , 0)
(4)
(5)
1
ORDERS
DESCRIPTION OF THE GROUPS
SUBGROUPS
ABE-SUBGROUPS
3,5
3,5
3 ,5
4,5
4,9,5
4 ,5+1
4 ,9+1,5+1
4
Cyclic(Z4 ) , Z22
6
8
Z6 , symmetric(Σ3 )
4,6
Z8 , octic(D8 ), quaternion(Q8 ) 4,10,6
9
10
12
14
15
16
18
20
21
22
24
25
26
27
28
Z4 × Z2 , Z32
Z9 , Z23
Z10 , dihedral(D10 )
Z12 , tetrahedral(A4 ), D12
Dicyclic(Dic3 ), Z22 × Z3
Z14 , D14
Z15
Z16 , Dic4 , D16 , (Q8 × Z2 )
Z8 × Z2 , Z24 , Z4 × Z22 , Z42
Modular group of order 16
Quasihedral of order 16
D8 o Z2
(Z4 × Z2 ) o Z2
G4,4
Q8 o Z2
Z18 , Z3 × Z6 , D18
(Z3 × Z3 ) o Z2 , Σ3 × Z3
Z20 , Z10 × Z2 , D20
Dic5 , metacyclic
Z21 , Z7 o Z3
Z22 , D22
Z24 ,Z2 × Z12 ,Z2 × Z2 × Z6
D8 × Z3 ,Q8 × Z3
Sl(2, 3), A4 × Z2
Σ4 , D24 , Dic6
Z2 × Z2 × Σ3 , Z2 × (Z3 o Z4 )
Z4 × Σ3 ,Z3 o Z8
(Z6 × Z2 ) o Z2
Z25 , Z25
Z26 , D26
Z27 , Z9 × Z3 , Z33
(Z3 × Z3 ) o Z3 , Z9 o Z3
Z28 , Z14 × Z2 , D28 , Dic7
The equations reduce as follows:
GENERATOR’S
(g , k) ∼ (k, −g )
(g1 , k1 )(g2 , k2 ) ∼ (g2 , k2 )(g1 , k1 )
8,16
8,16
8, 15
3,6
4,8
6,10,16
8,10
4,10
4
5,11,19,19
3,6
4,7
6,9,13
7,10
4,9
4
5,8,16,14
3 ,7
4,7+1
6 ,9+1, 13 +14+
7+1,10
4 ,9+1
2
4
5 ,8+4+ ,16+4+ ,14+10+
11,15,27,67
11
15
35
23
15
23
6,12,16
28,14
6,10,22
10,14
4,10
4,14
8,16,32
20,12
15,26
30,34,18
54,22
26,10
30
3,8
4,16
4,10,28
19,10
6,10,28,12
11,15,27,67
10
12
30
22
14
18
6,12,12
15,12
6,10,19
9,12
4,9
4,13
8,16,32
18,10
13,24
21,24,12
43,19
21,9
22
3,8
4,15
4,10,28
18,9
6,10,25,11
11 , 16 , 25 , 51
10+1
12+4+
28+
21+
10+7+
20+
6 ,14,12+
16+ ,13+13
6 ,10 ,19+
9+1, 12+3+
4 ,9+1
4 ,13+4+
8 ,16,30
18+ ,10+
13+ ,23+
21+3+ ,24+ ,12+
40+ ,19+
21+ ,9+
22+
3 ,11
4 ,15+
4 ,12,40
22+ ,10+
6 ,10,25+ ,11+
3
(g , k) ∼ (g , mg + k)
where m ∈ Z.
For the cyclic groups Zn , the number r Zn
coincides with the number of subgroups of Zn .
This is just a consequence of division
algorithm.
The first conjeture is: The number r G is the
number of subgroups of a finite group G .
false
k = q1 g + r1
(1)
−g = q2 r1 + r2
r1 = q3 r2 + r3
(2)
.
.
.
±rm−1 = qm+1 rm
with 0 ≤ |rm−1 | < ... < |r1 | < |k|.
(g , k) ∼ (g , r1 ) ∼ (r1 , −g ) ∼ (r1 , r2 ) ∼
∼ (r2 , −r1 ) ∼ (r2 , −r3 ) ∼ · · · ∼
(rm−1 , rm−2 ) ∼ (rm1 , −qrm−1 ) ∼
(rm−1 , 0)
(3)
(4)
(5)
ORDERS
DESCRIPTION OF THE GROUPS
SUBGROUPS
ABE-SUBGROUPS
GENERATOR’S
4
6
8
Cyclic(Z4 ), Z22
Z6 , symmetric(Σ3 )
Z8 , D8 , quaternion(Q8 )
Z4 × Z2 , Z32
Z9 , Z23
Z10 , D10
Z12 , tetrahedral(A4 ), D12
Dic3 , Z22 × Z3
Z14 , D14
Z15
Z16 , Dic4 , D16 , (Q8 × Z2 )
Z8 × Z2 , Z24 , Z4 × Z22 , Z42
Modular group of order 16
Quasihedral of order 16
D8 o Z2
(Z4 × Z2 ) o Z2
G4,4
Q8 o Z2
Z18 , Z3 × Z6 , D18
(Z3 × Z3 ) o Z2 , Σ3 × Z3
Z20 , Z10 × Z2 , D20
Dic5 , metacyclic
Z21 , Z7 o Z3
Z22 , D22
Z24 ,Z2 × Z12 ,Z2 × Z2 × Z6
D8 × Z3 ,Q8 × Z3
Sl(2, 3), A4 × Z2
Σ4 , D24 , Dic6
Z2 × Z2 × Σ3 , Z2 × (Z3 o Z4 )
Z4 × Σ3 ,Z3 o Z8
(Z6 × Z2 ) o Z2
Z25 , Z25
Z26 , D26
Z27 , Z9 × Z3 , Z33
(Z3 × Z3 ) o Z3 , Z9 o Z3
Z28 , Z14 × Z2 , D28 , Dic7
3,5
4,6
4,10,6
8,16
3,6
4,8
6,10,16
8,10
4,10
4
5,11,19,19
11,15,27,67
11
15
35
23
15
23
6,12,16
28,14
6,10,22
10,14
4,10
4,14
8,16,32
20,12
15,26
30,34,18
54,22
26,10
30
3,8
4,16
4,10,28
19,10
6,10,28,12
3,5
4,5
4, 9 ,5
8,16
3,6
4,7
6, 9 ,13
7,10
4, 9
4
5,8, 16 ,14
11,15,27,67
10
12
30
22
14
18
6,12, 12
15,12
6,10, 19
9,12
4,9
4, 13
8,16,32
18,10
13,24
21, 24 ,12
43,19
21,9
22
3,8
4, 15
4,10,28
18,9
6,10, 25 ,11
3,5
4,5+1
1
4, 9+1 ,5+1
8,15
3,7
4,7+1
2
6, 9+1 ,13+14+
7+1,10
4, 9+1
4
5,8+4+ , 16+4+ ,14+10+
11,16,25,51
10+1
12+4+
28+
3
21+
10+7+
20+
6,14, 12+
16+ ,13+13
4
6,10 , 19+
9+1, 12+3+
4,9+1
4, 13+4+
8,16,30
5
18+ ,10+
13+ ,23+
21+3+ , 24+ ,12+
40+ ,19+
21+ ,9+
6
22+
3,11
4, 15+
4,12,40
22+ ,10+
6,10, 25+ ,11+
9
10
12
14
15
16
18
20
21
22
24
25
26
27
28
r
G
G
G
G
= r1 + r2 + ... + rl
D
Second conjecture: The numbers r1 n and
Dic
r1 n are the number of abelian subgroups of
Dn and Dicn respectively.
true
Any subgroup of D2n is a subgroup of hr i or is
of the form hsr i , r m i for m = n/d with d a
divisor of n.
Similarly, for the dicyclic groups, Dicn , the
number of subgroups coincides with
τ (2n) + σ(n). To prove this we can use the
following exact sequence
1 → Z2 → Dicn → D2n → 1 .
The assignation is as follows:
Cyclic with generator a is sent to the
class of (a, 0).
The subgroup with two generators a
and b is sent to the class of (a, b).
ORDERS
DESCRIPTION OF THE GROUPS
SUBGROUPS
ABE-SUBGROUPS
GENERATOR’S
4
6
8
Cyclic(Z4 ), Z22
Z6 , symmetric(Σ3 )
Z8 , D8 , quaternion(Q8 )
Z4 × Z2 , Z32
Z9 , Z23
Z10 , D10
Z12 , tetrahedral(A4 ), D12
Dic3 , Z22 × Z3
Z14 , D14
Z15
Z16 , Dic4 , D16 , (Q8 × Z2 )
Z8 × Z2 , Z24 , Z4 × Z22 , Z42
Modular group of order 16
Quasihedral of order 16
D8 o Z2
(Z4 × Z2 ) o Z2
G4,4
Q8 o Z2
Z18 , Z3 × Z6 , D18
(Z3 × Z3 ) o Z2 , Σ3 × Z3
Z20 , Z10 × Z2 , D20
Dic5 , metacyclic
Z21 , Z7 o Z3
Z22 , D22
Z24 ,Z2 × Z12 ,Z2 × Z2 × Z6
D8 × Z3 ,Q8 × Z3
Sl(2, 3), A4 × Z2
Σ4 , D24 , Dic6
Z2 × Z2 × Σ3 , Z2 × (Z3 o Z4 )
Z4 × Σ3 ,Z3 o Z8
(Z6 × Z2 ) o Z2
Z25 , Z25
Z26 , D26
Z27 , Z9 × Z3 , Z33
(Z3 × Z3 ) o Z3 , Z9 o Z3
Z28 , Z14 × Z2 , D28 , Dic7
3,5
4,6
4,10,6
8,16
3,6
4,8
6,10,16
8,10
4,10
4
5,11,19,19
11,15,27,67
11
15
35
23
15
23
6,12,16
28,14
6,10,22
10,14
4,10
4,14
8,16,32
20,12
15,26
30,34,18
54,22
26,10
30
3,8
4,16
4,10,28
19,10
6,10,28,12
3,5
4,5
4, 9 ,5
8,16
3,6
4,7
6, 9 ,13
7 ,10
4, 9
4
5, 8 , 16 ,14
11,15,27,67
10
12
30
22
14
18
6,12, 12
15,12
6,10, 19
9 ,12
4,9
4, 13
8,16,32
18,10
13,24
21, 24 , 12
43,19
21,9
22
3,8
4, 15
4,10,28
18,9
6,10, 25 , 11
3,5
4,5+1
1
4, 9+1 ,5+1
8,15
3,7
4,7+1
2
6, 9+1 ,13+14+
7+1 ,10
4, 9+1
4
5, 8+4+ , 16+4+ ,14+10+
11,16,25,51
10+1
12+4+
28+
3
21+
10+7+
20+
6,14, 12+
16+ ,13+13
4
6,10 , 19+
9+1 , 12+3+
4,9+1
4, 13+4+
8,16,30
5
18+ ,10+
13+ ,23+
21+3+ , 24+ , 12+
40+ ,19+
6
21+ ,9+
22+
3,11
4, 15+
4,12,40
22+ ,10+
6,10, 25+ , 11+
9
10
12
14
15
16
18
20
21
22
24
25
26
27
28
r
G
G
G
G
= r1 + r2 + ... + rl
D
Second conjecture: The numbers r1 n and
Dic
r1 n are the number of abelian subgroups of
Dn and Dicn respectively.
true
Any subgroup of D2n is a subgroup of hr i or is
of the form hsr i , r m i for m = n/d with d a
divisor of n.
Similarly, for the dicyclic groups, Dicn , the
number of subgroups coincides with
τ (2n) + σ(n). To prove this we can use the
following exact sequence
1 → Z2 → Dicn → D2n → 1 .
The assignation is as follows:
Cyclic with generator a is sent to the
class of (a, 0).
The subgroup with two generators a
and b is sent to the class of (a, b).
Symmetric and alternating groups
Third conjecture: The numbers r1G is the number of abelian subgroups of
G.
false
ORDERS
2
6
24
120
720
5040
DESCRIPTION
Σ2
Σ3
Σ4
Σ5
Σ6
Σ7
SUBGROUPS
2
6
30
156
1455
11300
ABE-SUBGROUPS
2
5
21
87
612
3649
GENERATOR’S
2
5+1
21+3
87
592
3509
ORDERS
1
3
12
60
360
2520
DESCRIPTION
A2
A3
A4
A5
A6
A7
SUBGROUPS
1
2
10
59
501
3786
ABE-SUBGROUPS
1
2
9
37
207
1192
GENERATOR’S
1
2
9+1
37
216
1262
ORDERS
DESCRIPTION OF THE GROUPS
4
6
8
Cyclic(Z4 ), Z22
3,5
Z6 ,symmetric(Σ3 )
4,6
Z8 , octic(D8 ), quaternion(Q8 ) 4,10,6
Z4 × Z2 ,
9
10
12
14
15
16
Z32
Z9 , Z23
Z10 , dihedral(D10 )
Z12 , tetrahedral(A4 ), D12
Dicyclic(Dic3 ), Z22 × Z3
Z14 , D14
Z15
Z16 , Dic4 , D16 , (Q8 × Z2 )
Z2 , Z24 ,
18
20
21
22
24
25
26
27
28
Z22
Z42
Z8 ×
Z4 ×
,
Modular group of order 16
Quasihedral of order 16
D8 o Z2
(Z4 × Z2 ) o Z2
G4,4
Q8 o Z2
Z18 , Z3 × Z6 , D18
(Z3 × Z3 ) o Z2 , Σ3 × Z3
Z20 , Z10 × Z2 , D20
Dic5 , metacyclic
Z21 , Z7 o Z3
Z22 , D22
Z24 ,Z2 × Z12 ,Z2 × Z2 × Z6
D8 × Z3 ,Q8 × Z3
Sl(2, 3), A4 × Z2
Σ4 , D24 , Dic6
Z2 × Z2 × Σ3 , Z2 × (Z3 o Z4 )
Z4 × Σ3 ,Z3 o Z8
(Z6 × Z2 ) o Z2
Z25 , Z25
Z26 , D26
Z33
Z27 , Z9 × Z3 ,
(Z3 × Z3 ) o Z3 , Z9 o Z3
Z28 , Z14 × Z2 , D28 , Dic7
SUBGROUPS
ABE-SUBGROUPS
GENERATOR’S
3,5
4,5
4,9,5
3, 5
4,5+1
4,9+1,5+1
8,16
8,16
8, 15
3,6
4,8
6,10,16
8,10
4,10
4
5,11,19,19
3,6
4,7
6,9,13
7,10
4,9
4
5,8,16,14
3, 7
4,7+1
6,9+1,13+14+
7+1,10
4,9+1
4
5,8+4+ ,16+4+ ,14+10+
11,15,27,67
11
15
35
23
15
23
6,12,16
28,14
6,10,22
10,14
4,10
4,14
8,16,32
20,12
15,26
30,34,18
54,22
26,10
30
11,15,27,67
10
12
30
22
14
18
6,12,12
15,12
6,10,19
9,12
4,9
4,13
8,16,32
18,10
13,24
21,24,12
43,19
21,9
22
11,16,25, 51
10+1
12+4+
28+
21+
10+7+
20+
6,14,12+
16+ ,13+13
6,10 ,19+
9+1, 12+3+
4,9+1
4,13+4+
8,16,30
18+ ,10+
13+ ,23+
21+3+ ,24+ ,12+
40+ ,19+
21+ ,9+
22+
3,8
4,16
3,8
4,15
3, 11
4,15+
4,10,28
19,10
6,10,28,12
4,10,28
18,9
6,10,25,11
4,12, 40
22+ ,10+
6,10,25+ ,11+
1
There is the identity
n
Z
r p =
p 2n−1 + p n+1 − p n−1 + p 2 − p − 1
p2 − 1
n
2
The number r Z2 follows the sequence
2,5,15,51,187,...
3
Fourth conjecture: The number r Z2 is the
dimension of the universal embedding of the
symplectic dual polar space which writes
n
(2n +1)(2n−1 +1)
.
3
true
4
The correspondence
(g , k) ∼ (k, −g ) ↔
0
−1
(g , k) ∼ (g , mg + k) ↔
where m ∈ Z.
1
m
1
0
0
1
1
For a finite group G the number r1G is the number of orbits of the
action of Sl(2, Z) on the commuting elements
G (1) = {(g , k) : [k, g ] = 1}.
2
For a finite group G the number r2G is the number of orbits of the
action of the modular group Map(S2 ) on the elements
G (2) = {(g , k)(g 0 , k 0 ) : [k, g ][k 0 , g 0 ] = 1}.
3
...
4
For a finite group G the number rmG is the number of orbits of the
action of the modular group Map(Sm ) on the elements
G (m) = {(g1 , k1 )....(gm , km ) : [k1 , g1 ]...[km , gm ] = 1}.
5
For m = 3, what information do we obtain for the Ore’s conjecture?
6
We can not say nothing about the Thompson’s conjecture, since we
can prove that the conjugation classes of the labels associated to
separating curves do not change by the application of the equations.