Affiliation:
1. Alfréd Rényi Institute of Mathematics , Hungarian Academy of Sciences , Budapest , Hungary
2. Dipartimento di Matematica e Informatica “U. Dini” , Università degli Studi di Firenze , Firenze , Italy
Abstract
Abstract
Suppose that
w
=
w
(
x
1
,
…
,
x
n
)
w=w(x_{1},\ldots,x_{n})
is a word, i.e. an element of the free group
F
=
⟨
x
1
,
…
,
x
n
⟩
F=\langle x_{1},\ldots,x_{n}\rangle
.
The verbal subgroup
w
(
G
)
w(G)
of a group 𝐺 is the subgroup generated by the set
{
w
(
x
1
,
…
,
x
n
)
:
x
1
,
…
,
x
n
∈
G
}
\{w(x_{1},\ldots,x_{n}):x_{1},\ldots,x_{n}\in G\}
of all 𝑤-values in 𝐺.
Following J. González-Sánchez and B. Klopsch, a group 𝐺 is 𝑤-maximal if
|
H
:
w
(
H
)
|
<
|
G
:
w
(
G
)
|
\lvert H:w(H)\rvert<\lvert G:w(G)\rvert
for every
H
<
G
H<G
.
In this paper, we give new results on 𝑤-maximal groups, and study the weaker condition in which the previous inequality is not strict.
Some applications are given: for example, if a finite group has a solvable (resp. nilpotent) section of size 𝑛, then it has a solvable (resp. nilpotent) subgroup of size at least 𝑛.