In mathematics , in the field of group theory , especially in the study of p -groups and pro-p -groups , the concept of powerful p -groups plays an important role. They were introduced in (Lubotzky & Mann 1987 ), where a number of applications are given, including results on Schur multipliers . Powerful p -groups are used in the study of automorphisms of p -groups (Khukhro 1998 ), the solution of the restricted Burnside problem (Vaughan-Lee 1993 ), the classification of finite p -groups via the coclass conjectures (Leedham-Green & McKay 2002 ), and provided an excellent method of understanding analytic pro-p -groups (Dixon et al. 1991 ).
A finite p -group
G
{\displaystyle G}
is called powerful if the commutator subgroup
[
G
,
G
]
{\displaystyle [G,G]}
is contained in the subgroup
G
p
=
⟨ ⟨ -->
g
p
|
g
∈ ∈ -->
G
⟩ ⟩ -->
{\displaystyle G^{p}=\langle g^{p}|g\in G\rangle }
for odd
p
{\displaystyle p}
, or if
[
G
,
G
]
{\displaystyle [G,G]}
is contained in the subgroup
G
4
{\displaystyle G^{4}}
for
p
=
2
{\displaystyle p=2}
.
Properties of powerful p -groups
Powerful p -groups have many properties similar to abelian groups , and thus provide a good basis for studying p -groups. Every finite p -group can be expressed as a section of a powerful p -group.
Powerful p -groups are also useful in the study of pro-p groups as it provides a simple means for characterising p -adic analytic groups (groups that are manifolds over the p -adic numbers): A finitely generated pro-p group is p -adic analytic if and only if it contains an open normal subgroup that is powerful: this is a special case of a deep result of Michel Lazard (1965).
Some properties similar to abelian p -groups are: if
G
{\displaystyle G}
is a powerful p -group then:
The Frattini subgroup
Φ Φ -->
(
G
)
{\displaystyle \Phi (G)}
of
G
{\displaystyle G}
has the property
Φ Φ -->
(
G
)
=
G
p
.
{\displaystyle \Phi (G)=G^{p}.}
G
p
k
=
{
g
p
k
|
g
∈ ∈ -->
G
}
{\displaystyle G^{p^{k}}=\{g^{p^{k}}|g\in G\}}
for all
k
≥ ≥ -->
1.
{\displaystyle k\geq 1.}
That is, the group generated by
p
{\displaystyle p}
th powers is precisely the set of
p
{\displaystyle p}
th powers.
If
G
=
⟨ ⟨ -->
g
1
,
… … -->
,
g
d
⟩ ⟩ -->
{\displaystyle G=\langle g_{1},\ldots ,g_{d}\rangle }
then
G
p
k
=
⟨ ⟨ -->
g
1
p
k
,
… … -->
,
g
d
p
k
⟩ ⟩ -->
{\displaystyle G^{p^{k}}=\langle g_{1}^{p^{k}},\ldots ,g_{d}^{p^{k}}\rangle }
for all
k
≥ ≥ -->
1.
{\displaystyle k\geq 1.}
The
k
{\displaystyle k}
th entry of the lower central series of
G
{\displaystyle G}
has the property
γ γ -->
k
(
G
)
≤ ≤ -->
G
p
k
− − -->
1
{\displaystyle \gamma _{k}(G)\leq G^{p^{k-1}}}
for all
k
≥ ≥ -->
1.
{\displaystyle k\geq 1.}
Every quotient group of a powerful p -group is powerful.
The Prüfer rank of
G
{\displaystyle G}
is equal to the minimal number of generators of
G
.
{\displaystyle G.}
Some less abelian-like properties are: if
G
{\displaystyle G}
is a powerful p -group then:
G
p
k
{\displaystyle G^{p^{k}}}
is powerful.
Subgroups of
G
{\displaystyle G}
are not necessarily powerful.
References
Lazard, Michel (1965), Groupes analytiques p-adiques, Publ. Math. IHÉS 26 (1965), 389–603.
Dixon, J. D.; du Sautoy, M. P. F. ; Mann, A.; Segal, D. (1991), Analytic pro-p-groups , Cambridge University Press , ISBN 0-521-39580-1 , MR 1152800
Khukhro, E. I. (1998), p-automorphisms of finite p-groups , Cambridge University Press , doi :10.1017/CBO9780511526008 , ISBN 0-521-59717-X , MR 1615819
Leedham-Green, C. R. ; McKay, Susan (2002), The structure of groups of prime power order , London Mathematical Society Monographs. New Series, vol. 27, Oxford University Press , ISBN 978-0-19-853548-5 , MR 1918951
Lubotzky, Alexander ; Mann, Avinoam (1987), "Powerful p-groups. I. Finite Groups", J. Algebra , 105 (2): 484– 505, doi :10.1016/0021-8693(87)90211-0 , MR 0873681
Vaughan-Lee, Michael (1993), The restricted Burnside problem (2nd ed.), Oxford University Press , ISBN 0-19-853786-7 , MR 1364414