Page 127 - 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. 127
mož Moravec: Some Topics in the Theory of Finite Groups 115
classes of groups of order at most 2000, and 49, 487, 365, 422, or just over 99%, are groups
of order 1024. We mention here that Phillip Hall proved that the number of isomorphism
classes of groups of order p n is
p .2 n 3 +O (n 8/3 )
27
We will not prove this result. Instead we will derive some good upper and lower bounds
on the number of finite p -groups of given order. We refer to [2] for a wealth of further
estimates.
Preliminary results
Let r be a positive integer and Fr a free group on {x1, . . . , xr }. Denote
Gr = Fr /Frp2 γ2(Fr )p γ3(F ).
We identify xi with their images in Gr , so x1, . . . , xr generate Gr .
A finite p -group G is said to have Φ-class 2 if there exists a central elementary abelian
subgroup H of G such that G /H is elementary abelian. In other words, G is a central ex-
tension of an elementary abelian group by an elementary abelian group. Our first result
shows that every group of Φ-class 2 is a homomorphic image of some Gr :
Lemma 3.5.28 Let H be a group of Φ-class 2, and let y1, . . . , yr ∈ H . There is a homomor-
phism φ :Gr → H such that x φ = yi for all i = 1, . . . , r .
i
PROOF. As Fr is free there exists a unique homomorphism Fr → H with xi → yi . As
Frp2 γ2(Fr )p γ3(F ) is contained in the kernel of this map, we get the result.
Lemma 3.5.29 The group Gr is a finite p -group. The Frattini subgroup Frat(Gr ) is central
of order p r (r +1)/2 and index p r . Moreover, any automorphism α ∈ Aut(Gr ) that induces an
identity mapping on Gr / Frat(Gr ) fixes Frat(Gr ) pointwise.
PRO O F.[Sketch of proof ] The group G p γ2 (G r ) is a central elementary abelian p -subgroup
r
of Gr , and the quotient by it is also elementary abelian. Thus Gr is a p -group. Observe
that Frat(Gr ) is generated by x p and [x j , xi ], where 1 ≤ i < j ≤ r. It is straightforward but
i
technical to prove that this generating set is a minimal one, we skip the details. It follows
that Frat(Gr ) is central of order p r (r +1)/2 and index p r .
Now take α ∈ Aut(Gr ) that induces an identity mapping on Gr / Frat(Gr ). So there exist
h1,...,hr ∈ Frat(Gr ) such that x α = hi xi . Since Frat(Gr ) is central and Frat(G r )p = {1}, we
i
have
(x p )α = (x iα )p = (hi xi )p = h p x p = x p
i i i i
and
[x j , xi ]α = [x jα, x α ] = [h j x j , hi xi ] = [x j , xi ].
i
classes of groups of order at most 2000, and 49, 487, 365, 422, or just over 99%, are groups
of order 1024. We mention here that Phillip Hall proved that the number of isomorphism
classes of groups of order p n is
p .2 n 3 +O (n 8/3 )
27
We will not prove this result. Instead we will derive some good upper and lower bounds
on the number of finite p -groups of given order. We refer to [2] for a wealth of further
estimates.
Preliminary results
Let r be a positive integer and Fr a free group on {x1, . . . , xr }. Denote
Gr = Fr /Frp2 γ2(Fr )p γ3(F ).
We identify xi with their images in Gr , so x1, . . . , xr generate Gr .
A finite p -group G is said to have Φ-class 2 if there exists a central elementary abelian
subgroup H of G such that G /H is elementary abelian. In other words, G is a central ex-
tension of an elementary abelian group by an elementary abelian group. Our first result
shows that every group of Φ-class 2 is a homomorphic image of some Gr :
Lemma 3.5.28 Let H be a group of Φ-class 2, and let y1, . . . , yr ∈ H . There is a homomor-
phism φ :Gr → H such that x φ = yi for all i = 1, . . . , r .
i
PROOF. As Fr is free there exists a unique homomorphism Fr → H with xi → yi . As
Frp2 γ2(Fr )p γ3(F ) is contained in the kernel of this map, we get the result.
Lemma 3.5.29 The group Gr is a finite p -group. The Frattini subgroup Frat(Gr ) is central
of order p r (r +1)/2 and index p r . Moreover, any automorphism α ∈ Aut(Gr ) that induces an
identity mapping on Gr / Frat(Gr ) fixes Frat(Gr ) pointwise.
PRO O F.[Sketch of proof ] The group G p γ2 (G r ) is a central elementary abelian p -subgroup
r
of Gr , and the quotient by it is also elementary abelian. Thus Gr is a p -group. Observe
that Frat(Gr ) is generated by x p and [x j , xi ], where 1 ≤ i < j ≤ r. It is straightforward but
i
technical to prove that this generating set is a minimal one, we skip the details. It follows
that Frat(Gr ) is central of order p r (r +1)/2 and index p r .
Now take α ∈ Aut(Gr ) that induces an identity mapping on Gr / Frat(Gr ). So there exist
h1,...,hr ∈ Frat(Gr ) such that x α = hi xi . Since Frat(Gr ) is central and Frat(G r )p = {1}, we
i
have
(x p )α = (x iα )p = (hi xi )p = h p x p = x p
i i i i
and
[x j , xi ]α = [x jα, x α ] = [h j x j , hi xi ] = [x j , xi ].
i
