Page 129 - Ellingham, Mark, Mariusz Meszka, Primož Moravec, Enes Pasalic, 2014. 2014 PhD Summer School in Discrete Mathematics. Koper: University of Primorska Press. Famnit Lectures, 3.
P. 129
mož Moravec: Some Topics in the Theory of Finite Groups 117
orbits of AutGr on X . This gives the desired bound.
Proposition 3.5.32 yields roughly p x2y n3/2 groups with Frattini subgroup of index p xn
and order p y n . Maximizing the function z = x 2y /2 under the constraint x + y = 1 yields
the maximum value z = 2/27.
Theorem 3.5.33 The number f (p n ) of groups of order p n is at least
p 2 n 2 (n −6).
27
PROOF. We may assume n > 6. Define s = (n + 2(n mod 3))/3 and r = n − s . Then Propo-
sition 3.5.32 gives f (p n ) ≥ p r s (r +1)/2−r 2−s 2 ≥ p 2n2(n−6)/27.
An elementary upper bound
Let G be a group of order p n and let
G = G0 ≥ G1 ≥ · · · ≥ Gn−1 ≥ Gn = {1}
be its chief series. For each i choose g i ∈ Gi −1 − Gi . Then every g ∈ G may be written
uniquely in normal form g = g α1 · · · g αn , where αi ∈ {0, 1, . . . , p − 1}. Furthermore, g ∈ Gi
1 n
iff α1 = · · · = αi = 0.
Observe that, given 1 ≤ i < j ≤ n, we have that g p ∈ Gi and [g j , gi ] ∈ Gj . Hence we
i
may write these elements in normal form, that is,
g p = g βi ,i +1 · · · g βi ,n (3.1)
i n
i +1
and
[g j , g i ] = g γi ,j ,j +1 · · · g γi ,j ,n (3.2)
j +1 n
for some βi ,j , γi ,j ,k ∈ {0, 1, . . . , p − 1}. It is easy to see that the generators g 1, . . . , g n and
all the relations of the form (3.1) and (3.2) form a presentation for G (called a power
commutator presentation or polycyclic presentation). One has to prove that a product
of two elements in normal form can again be written in normal form. This can be done
using collection process described in [9].
We remark that GAP calls the groups given by power-commutator presentations pc
groups. Here is an example of how GAP prints out presentations of pc groups:
gap> PrintPcpPresentation(PcGroupToPcpGroup(DihedralGroup(16)));
g1^2 = id
g2^2 = g3
g3^2 = g4
g4^2 = id
g2 ^ g1 = g2 * g3 * g4
g3 ^ g1 = g3 * g4
orbits of AutGr on X . This gives the desired bound.
Proposition 3.5.32 yields roughly p x2y n3/2 groups with Frattini subgroup of index p xn
and order p y n . Maximizing the function z = x 2y /2 under the constraint x + y = 1 yields
the maximum value z = 2/27.
Theorem 3.5.33 The number f (p n ) of groups of order p n is at least
p 2 n 2 (n −6).
27
PROOF. We may assume n > 6. Define s = (n + 2(n mod 3))/3 and r = n − s . Then Propo-
sition 3.5.32 gives f (p n ) ≥ p r s (r +1)/2−r 2−s 2 ≥ p 2n2(n−6)/27.
An elementary upper bound
Let G be a group of order p n and let
G = G0 ≥ G1 ≥ · · · ≥ Gn−1 ≥ Gn = {1}
be its chief series. For each i choose g i ∈ Gi −1 − Gi . Then every g ∈ G may be written
uniquely in normal form g = g α1 · · · g αn , where αi ∈ {0, 1, . . . , p − 1}. Furthermore, g ∈ Gi
1 n
iff α1 = · · · = αi = 0.
Observe that, given 1 ≤ i < j ≤ n, we have that g p ∈ Gi and [g j , gi ] ∈ Gj . Hence we
i
may write these elements in normal form, that is,
g p = g βi ,i +1 · · · g βi ,n (3.1)
i n
i +1
and
[g j , g i ] = g γi ,j ,j +1 · · · g γi ,j ,n (3.2)
j +1 n
for some βi ,j , γi ,j ,k ∈ {0, 1, . . . , p − 1}. It is easy to see that the generators g 1, . . . , g n and
all the relations of the form (3.1) and (3.2) form a presentation for G (called a power
commutator presentation or polycyclic presentation). One has to prove that a product
of two elements in normal form can again be written in normal form. This can be done
using collection process described in [9].
We remark that GAP calls the groups given by power-commutator presentations pc
groups. Here is an example of how GAP prints out presentations of pc groups:
gap> PrintPcpPresentation(PcGroupToPcpGroup(DihedralGroup(16)));
g1^2 = id
g2^2 = g3
g3^2 = g4
g4^2 = id
g2 ^ g1 = g2 * g3 * g4
g3 ^ g1 = g3 * g4
