Torsionfree Varieties of Metabelian Groups
L. G. Kovdcs and M. E Newman
Abstract. It is proved here that the free groups of the variety 01,21, A 21
usual,
2 a r 21
e denotes
t o r the
s i variety
o n f of
r eabelian
e . groups,A 21, thes variety of abelian groups of exponent
dividing s, and 01, the variety of nilpotent groups of class at most c.
1991 Mathematics Subject Classification: 20E10
We call a variety of groups torsionfree if its free groups are torsionfree. Apadfrom
this, we follow the notation and terminology of Hanna Neumann's book [4]. Recent
work by Samuel M. Vovsi and the first author [3] on the growth of varieties of groups
depends (among other things) on results of J. R. J. Groves [2]. In turn, these make
use of Ike classification of the torsionfree varieties of metabelian groups, Theorem
6.1.2 in R. A. Bryce [1]. All but one step of the proof of this classification was given
in Appendix I of [1], but our 'forthcoming' paper which would have contained the
missing step was never written. In view of the renewed interest, it seems desirable to
place the missing step on record.
Theorem (6.1.2 in [1]). The varieties of groups 9 7
free
, -. and join-irreducible. Every torsionfree proper subvariety of 91
torsionfree join-irreducibles.
2expressed
9 1c a n as an
b irredundant
e
u n join
i qofusome
e of
l these
y
s A
2 1
2 All (joins
C mentioned
,
here
S are the joins
> of finitely many join-irreducibles. The
uniqueness
important
part of the Theorem. It implies (by very
1
) claim is aaparticularly
r
e
simple
t o and
r general
s i lattice-theoretic
o
n
- considerations) that one join of join-irreducibles,
V,U , is contained in another, V
it,
j 9c 9:1J.
3 i In ,particular,
i f a join of join-irreducibles is irredundant if and only if its components
a n are
d pairwise incomparable. It is also well-known that comparability is easy
to
o settle
n here:
l yg t
with
these
ci
f points in mind that one speaks of a 'classification'. We conclude the paper
with
2t t a short
o argument which justifies these points independently: instead of appealing
to
se theAuniqueness
a 2 ct claim,
h it actually implies that claim.
2i
C
O
i
t
hfrom e
r
Offprint
cInfinite
e
Groups 94, Eds.: de Giov anni/ Newell
,©
i %
by Walter de
s Gruyter & Co., Berlin • New York 1995
sa
t/
tr ux /'
126
L
.
G. Kovdcs and M. E Newman
The outstanding step in the proof of the theorem itself is to show the first of its
claims, namely that each 01„21, A 20 is torsionfree. The key point of the proof is the
following.
Lemma. Let G be a free group of the variety 01, A 21
let
2 fmrbe
e an
e l integer
y g with
e n1 <
e mr <a n,t and
e ddenote by H the subgroup of G generated by
the
subgroup G' and the elements g
b commutator
y
factor
m
group
H
g
i
, I H' is torsionfree. ,
+
,
g
g1,
n
Proof
A ll we really need to show is that G ' / H' is torsionfree. We shall be working
with
basic
.,
T commutators
h e n with reference to the given free generating set of G. As is
well
known
(see
t
h
e 36.32 in [4]), the left-normed basic commutators of weight at least 2
and
at
most
c
group. Let X denote the union of this
c o m form
m ua basis
t of
a G'tas free
o abelian
r
set with I gJI m < j < n 1. Clearly, H is just the subgroup generated by X. Let I' be
the set of those left normed basic commutators of weight at least 2 whose last entry
has subscript larger than m: thus a typical element y of I' has the form [ g
t i t > 2 and i
with
when
t, >, t =m2g then
.i because y is basic. Thus 1
We
claim
1
C
H
'
. the subgroup (1
t e r e that
H
centralized
1
by
G', so what we have to show to prove this claim is that if y E I ' and
[] g t
e n[ye, gi]
r a Et (Y).
e d If i
i >) mgthen
i2
, t<b << iy j
in
t, a
t Ymetabelian
h e n group the order of the entries of a left-normed commutator is irrelevant
beyond
(see 34.51 in [4]). Finally, if i < i2 then i 2 > j > m
tg h[ i e nythesfirst two places
,
and
m
<
<
i2
<
•
•
•
<
i
[t gtyn i o j r
m
a
l
t,_ giibiI em c<sa <un sn l.e In this case of course [y, g
yg
commutators
at
.s i
s weight
c least t +1, all entries of all the basic commutators involved
i i bHi a i of
b
coming
sn i s{ b
cg f ,e
]] ] aOac afrom
isil[yww
, cgr i i ro t o t i e u
s
tn
t r a
we claim
that if u, v E X then [u, vi E (V). If u, v E G ' then [u, v] -,- 1
.seta
la
meNext
<
l
l
s sn e
while
G
]i pa<Y ifrn u, o)v n
jtn
. d' then
u either
c
t[u, v] or [u, v ]
-T
1
l
i
e
s
i
n
V
.
the
case
of
u
[
g
,
1
,
,
g
+
ef
o
nl oi di h
sit
[u,
v]
o
is
basic
as
written
i
t
and
so
lies
in Y; otherwise i
1
n
eblss
a i
s
i
c
tX
[u,
v]
E
(Y)
follows
by
the
previous
paragraph.
t w
>u i ltf h
j fit c tr> e s
m
]sa
e
b
s
:i eWenhave
tliasn
o o proved
u the subset Y of H'
E generates a normal subgroup in H which
>
, that
i e thes2commutator
b
contains
of
each
pair
of
elements from the generating set X of H:
Y
e
a
a
l
n
d
vd
to
e
sy( n ih
thus
=
H'.
Since
Y
is
a
subset
of
a
basis
of the free abelian group G', it follows
t•sfcY (Y)
w
h
i
e
t
n
h
Y Gee'/=H' is also free abelian.
d
that
g
[a
b
fi) . n a
fsys. I
i i
iProoft of
the first
w
t claim
h of the Theorem. It suffices to deal with noncyclic relatively free
c,w
f niof finite
ogroups
rank.
Let F be a noncyclic absolutely free group of rank m, and let
m
cg
m .
rw
io
A
be
the
verbal
subgroup
2t,(G). By Schreier's Theorem, A is also absolutely free of
Im
jth
u
t f
t rank
finite
and
its
rank
is
greater
than m: denote this rank by n. Set N = A '
n
]e
a
t
oo
e1
w
rb
n
n0 1 ,s( A )e a n d
ilb
o
t
tin
g
u
>
<
tt2
e
jio
i,
tn
h
Torsionfree Varieties of Metabelian Groups
1
2
7
G A I N : then G is an (T, A 2 1
2
abelian
of rank m (because it is a subgroup of finite index in the free abelian group
)F-IfFr 'e of
e rank
g m),
r osouA/A'
p splits over F'IA', and F'/A ' is free abelian of rank n - m.
Choose
a preimage {gi , , g
o
f a basis for
r Aalf '', nand choose
k
Similarly,
choose
g
m
n
.
m
A
}T +f o rh t he a t
b a s i s
known
l1
, o 31.25
g
iq , un(see
Ai [4]),eIthisnimplies
N
. that Igi , , g
t in
n
32.1,
iter,i and
it
A n41.33 /in [4]) generates it freely. We may therefore set H , F' I N, note
IN
that
g
H'
e
n
,
e
Ir N
N
a t, e and
s apply the
G Lemma to conclude that T''/F''N is torsionfree.
F
'
/
F
Of
is/ tcourse
s oethenl F/T'''N
f , is also atorsionfree.
n Since
d T'''N = f'''9 /,(A) by the definition
of
since
01,(A)
=
(
F
)
,
we
have
that f ' ' ' N = ( T A , A 2 1
(ist N and
s
e
e
h
as
2
completes
the
proof
of
the
Theorem.
tf
a
r
e
)b
te( F ) . hT h i s
e As promised in the introduction, we close with a simple proof of the fact that
a
si
m
(1)
01
ia
g
c
cannot hold unless for some
i we have sls(i) and c < c(i).
9
se
is
fo Suppose that (1) holds.
A
By
Dirichlet's
Theorem,
there are infinitely many primes p such that st(p 1 ) .
f
o
2
For
such
a
p,
in
the
holomorph
of a group of order p one can find an element g of
rI
g
1
order
p +and an element h
m
F
2 of order s. The subgroup (g, h) is in 2 1
in
p
'1 T,21, A 21
C
2
V
2
t
/,
[x, y
.sA
h e ni c e t ( 1 )a
l s o
, a n d
(
C
where
i the sequence t (1), , t (n) has c(i) 1 terms
'g o nn s,------i E (c (i) + 1) 0and for each
,
d
equal
is a law in each of the T, (0 2 1
.n e rto s(i). This commutator
1
•
ystT
e that it is notc a law in (g , h) unless some s(i) is divisible by s. This proves
I = hh shows
•
that
lh
(i),
bf ut ist at least
s eone
t (ti with
i n sgIs (i).
x
m e there
•
Suppose
now
that
c(i)
<
c
whenever
s
Is(i): we shall show that this leads to a
-e
=
g
i
,
o
y common multiple of the s(i) that are divisible by
contradiction.
Let
s'
denote
the
least
)
n o r m
d
n
Q validt if we replace all the corresponding join components
s.
The inclusion (1) remains
e
Iu d
l
o
on
the right hand side byt M-c-124
cg
o o
s
loss
of generality that s divides s )(1) but does not divide any of the other s(i), and that
1
m
ic(1) <mc.
(
]
.u I nt s tae a d
ot f
, Choose p large — say,
ct ho a r n g i n ) g so that also p > c + E (c (i) + s
,such( ithat, in the holomorph considered above, the conjugate g
n o t a t i A
o n
g
power
gm.
The
wreath
product
ofs order p is nilpotent of class p
h
)
i
)
s
.
L
t
h
e
e
t
s
m
%
a m W eof
b twoegroups
a
,
e
2
(which
of order s which acts trivially on
tw a h is larger
n ee thanic), nand
tit has
e angautomorphism
e r
n
)
the
top
group
and
mth
poweringly
on
the
base
group.
Let P , W/01,(W); let g be the
a
s
s
u
e
image
in
P
of
a
generator
of
one
of
the
coordinate
subgroups
of W, and k the image
m
e
rof a generator of the top group. It follows that P has an automorphism which sends
w
i
t
h
a
g
and fixes
o to gm u
t k. Let G be the semidirect product of P by (h), with h acting on P
tas the automorphism just described. The normal closure of g is abelian (of order pc),
e
s
G
m
o
d
,
128
L
.
G. Kovdcs and M. E Newman
and the factor group over that is cyclic (of order ps), so G E 0 1
to
c see that a nontrivial element of the normal closure of g and a nontrivial element of
2 1 ,can A
2 0 . Consider
I
t the left-normed
i
s commutator
(h)
never commute.
e
a
s
y
[
x
.
1
(
1
)
,
x
c
S
,
(
1
)
y
t
(
1
)
where this time n = E
i l c(i ) + 1 terms equal to s (i). This commutator is also a law in each of the 0 1
has
c( c setting
but
( i ) ±
(1) 2 t )
xi = g, x2 = • • • = x
sa
n
d
c in=G. The
k ,desired contradiction has been reached and the
(fshows
i ) , othat itr is not a law
y
=
e
a complete.
c
proof is
h
h
i
1
References
t
h
e[1] Bryce, R. A., Metabelian groups and varieties. Philos. Trans. Roy. Soc. London Ser. A
s 266
e (1970),
q 281-355.
u
e
n
[21 Groves, J. R. J., Varieties of soluble groups and a dichotomy of P. Hall. Bull. Austral.
c
e
Math. Soc. 5 (1971), 391-410.
t
[31
Kovdcs,
L. O., and Vovsi, S. M., Growth of varieties of groups and group representations,
,
and
the
Gel'
fand—Kirillov dimension. J. Algebra. To appear.
t
([41 Neumann, H., Varieties of groups. Springer-Verlag, Berlin Heidelberg New York 1967.
n
)
,
y
t
(
n
)
1