In mathematics, specifically group theory, finite groups of prime power order
pn
p
n\ge0
The p-group generation algorithm by M. F. Newman[1] and E. A. O'Brien[2] [3] is a recursive process for constructing the descendant treeof an assigned finite p-group which is taken as the root of the tree.
For a finite p-group
G
G
(Pj(G))j\ge
G
(1) P0(G):=G
Pj(G):=\lbrackPj-1(G),G\rbrack ⋅ Pj-1(G)p
j\ge1
Since any non-trivial finite p-group
G>1
c\ge1
Pc-1(G)>Pc(G)=1
clp(G):=c
G
1
clp(1)=0
G
clp(G):=min\lbracec\ge0\midPc(G)=1\rbrace
The complete lower p-central series of
G
(2) G=P0(G)>\Phi(G)=P1(G)>P2(G)> … >Pc-1(G)>Pc(G)=1
since
P1(G)=\lbrackP0(G),G\rbrack ⋅
p=\lbrack | |
P | |
0(G) |
G,G\rbrack ⋅ Gp=\Phi(G)
G
For the convenience of the reader and for pointing out the shifted numeration, we recall thatthe (usual) lower central series of
G
(\gammaj(G))j\ge
G
(3) \gamma1(G):=G
\gammaj(G):=\lbrack\gammaj-1(G),G\rbrack
j\ge2
As above, for any non-trivial finite p-group
G>1
c\ge1
\gammac(G)>\gammac+1(G)=1
cl(G):=c
G
c+1
G
1
cl(1)=0
The complete lower central series of
G
(4)
\prime | |
G=\gamma | |
1(G)>G |
=\gamma2(G)>\gamma3(G)> … >\gammac(G)>\gammac+1(G)=1
since
\gamma2(G)=\lbrack\gamma1(G),G\rbrack=\lbrackG,G\rbrack=G\prime
G
The following Rules should be remembered for the exponent-p class:
Let
G
cl(G)\leclp(G)
\gammaj(G)
Pj(G)
\vartheta\inHom(G,\tilde{G})
\tilde{G}
\vartheta(Pj(G))=Pj(\vartheta(G))
j\ge0
c\ge0
N\triangleleftG
clp(G/N)=c
Pc(G)\leN
c\ge0
clp(G)=c
clp(G/Pk(G))=min(k,c)
k\ge0
clp(G/Pk(G))=k
0\lek\lec
The parent
\pi(G)
G>1
clp(G)=c\ge1
\pi(G):=G/Pc-1(G)
G
Pc-1(G)>1
G
G
\pi(G)
clp(G)=clp(\pi(G))+1
A descendant tree is a hierarchical structurefor visualizing parent-descendant relationsbetween isomorphism classes of finite p-groups.The vertices of a descendant tree are isomorphism classes of finite p-groups.However, a vertex will always be labelled by selecting a representative of the corresponding isomorphism class.Whenever a vertex
\pi(G)
G
G\to\pi(G)
\pi:G\to\pi(G)
\pi(G)=G/Pc-1(G)
In a descendant tree, the concepts of parents and immediate descendants can be generalized.A vertex
R
P
P
R
R
P
(5) R=Q0\toQ1\to … \toQm-1\toQm=P
m\ge1
of directed edges from
R
P
j | |
Q | |
j=\pi |
(R)
R
0\lej\lem
(6) R=\pi0(R)\to\pi1(R)\to … \to\pim-1(R)\to\pim(R)=P
m\ge1
They can also be viewed as the successive quotients
Qj=R/Pc-j(R)
c-j
R
R
clp(R)=c\gem
(7) R\simeqR/Pc(R)\toR/Pc-1(R)\to … \toR/Pc+1-m(R)\toR/Pc-m(R)\simeqP
c\gem\ge1
In particular, every non-trivial finite p-group
G>1
c=clp(G)
(8) G\simeqG/1=G/Pc(G)\to\pi(G)=G/Pc-1
2(G)=G/P | |
(G)\to\pi | |
c-2 |
(G)\to …
… \to\pic-1
c(G)=G/P | |
(G)=G/P | |
0(G)=G/G\simeq |
1
ending in the trivial group
\pic(G)=1
G
\pic-1(G)=G/P1(G)\simeq
d | |
C | |
p |
d=d(G)
d(G)=\dim | |
Fp |
1(G,F | |
(H | |
p)) |
G
Generally, the descendant tree
l{T}(G)
G
G
G
l{T}(1)
1
1
d\ge1
p
Let
G
d
G
G\ast
G
G
We can certainly find a presentation of
G
(9) 1\longrightarrowR\longrightarrowF\longrightarrowG\longrightarrow1
where
F
d
\vartheta: F\longrightarrowG
R:=\ker(\vartheta)
R\triangleleftF
F
G\simeqF/R
r\inR
f\inF
f-1rf\inR
\lbrackr,f\rbrack=r-1f-1rf\inR
R
R\ast:=\lbrackR,F\rbrack ⋅ Rp
R
R/R\ast
G
(10) \lbrackR,R\rbrack ⋅ Rp\le\lbrackR,F\rbrack ⋅ Rp=R\ast
Now we can define the p-covering group of
G
(11) G\ast:=F/R\ast
and the exact sequence
(12) 1\longrightarrowR/R\ast\longrightarrowF/R\ast\longrightarrowF/R\longrightarrow1
shows that
G\ast
G
(13)
\mu(G):=\dim | |
Fp |
(R/R\ast)
the p-multiplicator rank of
G
Let us assume now that the assigned finite p-group
G\simeqF/R
clp(G)=c
R\triangleleftF
clp(F/R)=c
Pc(F)\leR
G
(14)
\ast | |
P | |
c(G |
)=Pc(F) ⋅ R\ast/R\ast\leR/R\ast
as a subgroup of the p-multiplicator.Consequently, the nuclear rank
(15)
\nu(G):=\dim | |
Fp |
\ast | |
(P | |
c(G |
))\le\mu(G)
of
G
As before, let
G
d
Proposition.Any p-elementary abelian central extension
(16) 1\toZ\toH\toG\to1
of
G
Z\le\zeta1(H)
d(H)=d(G)=d
G\ast
G
For the proof click show on the right hand side.
The reason is that, since
d(H)=d(G)=d
\psi: F\toH
\vartheta=\omega\circ\psi
\omega: H\toH/Z\simeqG
R=\ker(\vartheta)=\ker(\omega\circ\psi)=(\omega\circ\psi)-1(1)=\psi-1(\omega-1(1))=\psi-1(Z)
and thus
\psi(R)=\psi(\psi-1(Z))=Z
\psi(Rp)=Zp=1
Z
\psi(\lbrackR,F\rbrack)=\lbrackZ,H\rbrack=1
Z
\psi(R\ast)=\psi(\lbrackR,F\rbrack ⋅ Rp)=1
\psi
\psi\ast: G\ast\toH
H\simeqG\ast/\ker(\psi\ast)
In particular, an immediate descendant
H
G
(17) 1\toPc-1(H)\toH\toG\to1
of
G
1=Pc(H)=\lbrackPc-1(H),H\rbrack ⋅ Pc-1(H)p
Pc-1(H)p=1
Pc-1(H)\le\zeta1(H)
where
c=clp(H)
Definition.A subgroup
M/R\ast\leR/R\ast
G
M/R\ast=\ker(\psi\ast)
\psi\ast: G\ast\toH
H
G
An equivalent characterization is that
1<M/R\ast<R/R\ast
(18) (M/R\ast) ⋅ (Pc(F) ⋅ R\ast/R\ast)=R/R\ast
Therefore, the first part of our goal to compile a list of all immediate descendants of
G
R/R\ast
\ast | |
P | |
c(G |
)=Pc(F) ⋅ R\ast/R\ast
c=clp(G)
(19) \lbraceF/M \mid M/R\ast\leR/R\astisallowable\rbrace
where
G\ast/(M/R\ast)=(F/R\ast)/(M/R\ast)\simeqF/M
F/M1\simeqF/M2
Two allowable subgroups
\ast | |
M | |
1/R |
\ast | |
M | |
2/R |
F/M1\simeqF/M2
G
Such an isomorphism
\varphi: F/M1\toF/M2
G=F/R
c=clp(G)
\varphi(R/M1)=\varphi(Pc(F/M1))=Pc(\varphi(F/M1))=Pc(F/M2)=R/M2
\alpha\inAut(G)
G
\alpha\ast\inAut(G\ast)
G\ast=F/R\ast
G
\alpha\ast
R/R\ast
G
\alpha
Since
\alpha\ast(M/R\ast) ⋅
\ast | |
P | |
c(F/R |
)=\alpha\ast\lbrackM/R\ast ⋅
\ast | |
P | |
c(F/R |
)\rbrack=\alpha\ast(R/R\ast)=R/R\ast
\alpha\ast\inAut(G\ast)
\alpha\prime
M/R\ast\leR/R\ast
P:=\langle\alpha\prime\mid\alpha\inAut(G)\rangle
G
Aut(G)\toP
\alpha\mapsto\alpha\prime
M/R\ast\leR/R\ast
P
Eventually, our goal to compile a list
\lbraceF/Mi\mid1\lei\leN\rbrace
G
\ast | |
M | |
i/R |
N
R/R\ast
P
A finite p-group
G
\nu(G)
G
G
G
\nu(G)=0
G
\nu(G)\ge1
G=F/R
\nu=\nu(G)
1\les\le\nu
(R/R\ast:M/R\ast)=ps
M/R\ast
R/R\ast
G
\vertG\vert=pn
s
\#(F/M)=(F/R\ast:M/R\ast)=(F/R\ast:R/R\ast) ⋅ (R/R\ast:M/R\ast)
=\#(F/R) ⋅ ps=\vertG\vert ⋅ ps=pn ⋅ ps=pn+s
For the related phenomenon of multifurcation of a descendant tree at a vertex
G
\nu(G)\ge2
The p-group generation algorithm provides the flexibility to restrict the construction of immediate descendants to those of a single fixed step size
1\les\le\nu
We denote the number of all immediate descendants, resp. immediate descendants of step size
s
G
N
Ns
\nuN | |
N=\sum | |
s |
0\leCs\leNs
(N1/C1;\ldots;N\nu/C\nu)
First, let
p=3
We begin with groups having abelianization of type
(3,3)
\langle27,3\rangle
1
\nu=2
\mu=4
(4/1;7/5)
N=11
\langle243,3\rangle=\langle27,3\rangle-\#2;1
2
\nu=2
\mu=4
(10/6;15/15)
N=25
\langle729,40\rangle=\langle243,3\rangle-\#1;7
\nu=2
\mu=5
(16/2;27/4)
N=43
In contrast, groups with abelianization of type
(3,3,3)
\langle81,12\rangle
2
\nu=2
\mu=7
(10/2;100/50)
N=110
\langle243,37\rangle
3
\nu=5
\mu=9
(35/3;2783/186;81711/10202;350652/202266;\ldots)
N>4 ⋅ 105
\langle729,122\rangle
4
\nu=8
\mu=11
(45/3;117919/1377;\ldots)
N>105
Next, let
p=5
Corresponding groups with abelianization of type
(5,5)
p=3
\langle125,3\rangle
1
\nu=2
\mu=4
(4/1;12/6)
N=16
\langle3125,3\rangle=\langle125,3\rangle-\#2;1
2
\nu=3
\mu=5
(8/3;61/61;47/47)
N=116
Via the isomorphism
Q/Z\to\muinfty
n | \mapsto\exp\left( | |
d |
n | |
d |
⋅ 2\pii\right)
Q/Z=\left\lbrace | n |
d |
⋅ Z\midd\ge1, 0\len\led-1\right\rbrace
\muinfty=\lbracez\inC\midzd=1forsomeintegerd\ge1\rbrace
Let
p
G
G=F/R
M(G):=H2(G,Q/Z)
G
Q/Z
G
M(G)=(R\cap\lbrackF,F\rbrack)/\lbrackF,R\rbrack
I. R. Shafarevich[4] has proved that the difference between the relation rank
r(G)=\dim | |
Fp |
2(G,F | |
(H | |
p)) |
G
d(G)=\dim | |
Fp |
1(G,F | |
(H | |
p)) |
G
G
r(G)-d(G)=d(M(G))
N. Boston and H. Nover[5] have shown that
\mu(Gj)-\nu(Gj)\ler(G)
Gj:=G/Pj(G)
clp(Gj)=j
j\ge0
G
G/G\prime
Furthermore, J. Blackhurst (in the appendix On the nucleus of certain p-groups of a paper by N. Boston, M. R. Bush and F. Hajir[6])has proved that a non-cyclic finite p-group
G
M(G)
l{T}(1)
1
M(G)=1
⇒
\nu(G)=0
G
r(G)=d(G)
r(G)-d(G)=0=d(M(G))
M(G)=1
l{T}(1)
G
r(G)=d(G)+1
r(G)-d(G)=1=d(M(G))
M(G)