Some properties of nilpotent groups
Property S, a finiteness property which can hold in infinite groups, was introduced by Stallings and others and shown to hold in free groups. In [2] it was shown to hold in nilpotent groups as a consequence of a technical result of Mal'cev. In that paper this technical result was dubbed propert...
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Інститут прикладної математики і механіки НАН України
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| Цитувати: | Some properties of nilpotent groups / A.M. Gaglione, S. Lipschutz, D. Spellman // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 4. — С. 66–77. — Бібліогр.: 8 назв. — англ. |
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irk-123456789-1545992019-06-16T01:26:29Z Some properties of nilpotent groups Gaglione, A.M. Lipschutz, S. Spellman, D. Property S, a finiteness property which can hold in infinite groups, was introduced by Stallings and others and shown to hold in free groups. In [2] it was shown to hold in nilpotent groups as a consequence of a technical result of Mal'cev. In that paper this technical result was dubbed property R. Hence, more generally, any property R group satisfies property S. In [7] it was shown that property R implies the following (labeled there weak property R) for a group G: If G₀ is any subgroup in G and G₀* is any homomorphic image of G₀, then the set of torsion elements in G₀* forms a locally finite subgroup. It was left as an open question in [7] whether weak property R is equivalent to property R. In this paper we give an explicit counterexample thereby proving that weak property R is strictly weaker than property R. 2009 Article Some properties of nilpotent groups / A.M. Gaglione, S. Lipschutz, D. Spellman // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 4. — С. 66–77. — Бібліогр.: 8 назв. — англ. 1726-3255 2000 Mathematics Subject Classification:20F18,20F05,20F24,16D10. http://dspace.nbuv.gov.ua/handle/123456789/154599 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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Property S, a finiteness property which can hold in infinite groups, was introduced by Stallings and others and shown to hold in free groups. In [2] it was shown to hold in nilpotent groups as a consequence of a technical result of Mal'cev. In that paper this technical result was dubbed property R. Hence, more generally, any property R group satisfies property S. In [7] it was shown that property R implies the following (labeled there weak property R) for a group G: If G₀ is any subgroup in G and G₀* is any homomorphic image of G₀, then the set of torsion elements in G₀* forms a locally finite subgroup. It was left as an open question in [7] whether weak property R is equivalent to property R. In this paper we give an explicit counterexample thereby proving that weak property R is strictly weaker than property R. |
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Gaglione, A.M. Lipschutz, S. Spellman, D. |
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Gaglione, A.M. Lipschutz, S. Spellman, D. Some properties of nilpotent groups Algebra and Discrete Mathematics |
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Gaglione, A.M. Lipschutz, S. Spellman, D. |
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Gaglione, A.M. |
| title |
Some properties of nilpotent groups |
| title_short |
Some properties of nilpotent groups |
| title_full |
Some properties of nilpotent groups |
| title_fullStr |
Some properties of nilpotent groups |
| title_full_unstemmed |
Some properties of nilpotent groups |
| title_sort |
some properties of nilpotent groups |
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Інститут прикладної математики і механіки НАН України |
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2009 |
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http://dspace.nbuv.gov.ua/handle/123456789/154599 |
| citation_txt |
Some properties of nilpotent groups / A.M. Gaglione, S. Lipschutz, D. Spellman // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 4. — С. 66–77. — Бібліогр.: 8 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT gaglioneam somepropertiesofnilpotentgroups AT lipschutzs somepropertiesofnilpotentgroups AT spellmand somepropertiesofnilpotentgroups |
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2025-07-14T06:38:57Z |
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1837603370194436096 |
| fulltext |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Number 4. (2009). pp. 66 – 77
c⃝ Journal “Algebra and Discrete Mathematics”
Some properties of nilpotent groups
Anthony M. Gaglione, Seymour Lipschutz,
Dennis Spellman
Communicated by I. Ya. Subbotin
Dedicated to Professor Leonid A. Kurdachenko on his 60tℎ birthday
Abstract. Property S, a finiteness property which can hold in
infinite groups, was introduced by Stallings and others and shown
to hold in free groups. In [2] it was shown to hold in nilpotent
groups as a consequence of a technical result of Mal’cev. In that
paper this technical result was dubbed property R. Hence, more
generally, any property R group satisfies property S. In [7] it was
shown that property R implies the following (labeled there weak
property R) for a group G:
If G0 is any subgroup in G and G∗
0
is any homomorphic
image of G0, then the set of torsion elements in G∗
0
forms
a locally finite subgroup.
It was left as an open question in [7] whether weak property R is
equivalent to property R. In this paper we give an explicit coun-
terexample thereby proving that weak property R is strictly weaker
than property R.
1. An alphabet soup of properties
In this paper, we use the following notation. If G is a group and S ⊆ G,
then ⟨S⟩ denotes the subgroup of G generated by the elements of S. Also
⟨...; ...⟩ indicates a description of a group in terms of generators and
relations.
2000 Mathematics Subject Classification: 20F18,20F05,20F24,16D10.
Key words and phrases: Property S, Property R, commensurable, variety of
groups, closure operator.
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.A. M. Gaglione, S. Lipschutz, D. Spellman 67
Definition 1.1. Subgroups A and B of a group G are commensurable
provided has A ∩B finite index in each of A and B.
Definition 1.2. The group G satisfies property S provided, whenever
A and B are finitely generated commensurable subgroups, A∩B has finite
index in ⟨A,B⟩.
Below are examples of groups which violate property S. Let n > 1 be
an integer.
Example 1.3. The cyclically pinched one-relator group G = ⟨a, b; an =
bn⟩ . Let an = z = bn so that z is central in the torsion free group G.
Let A = ⟨a⟩ and B = ⟨b⟩. Then A∩B = ⟨z⟩ has finite index in each of A
and B but ⟨A,B⟩ /⟨z⟩ is the free product ⟨a; an = 1⟩ ∗ ⟨b; bn = 1⟩ which
is infinite.
Example 1.4. Suppose is n odd and sufficiently large (e.g., n ≥ 667
will do) so that the rank 2 free group F2(ℬn) in the Burnside variety of
exponent n is infinite. Suppose {a, b} freely generates F2(ℬn) relative to
ℬn. Let A = ⟨a⟩ and B = ⟨b⟩. Then A∩B = {1} has finite index in each
of A and B. But ⟨A,B⟩ /{1} ∼= G/{1} ∼= G is infinite.
Example 1.5. Let G = ⟨a, b; an = bn = 1⟩ be the free product ⟨a; an =
1⟩ ∗ ⟨b; bn = 1⟩. Let A = ⟨a⟩ and B = ⟨b⟩. Then A ∩ B = {1} has finite
index in each of A and B but ⟨A,B⟩ /{1} ∼= G/{1} ∼= G is infinite.
Theorem 1.6. Free groups satisfy property S.
For a proof see, for example, [2]. Note that, since there are examples
of groups which violate property S, property S is not, in general, preserved
in homomorphic images.
Now we go on an orgy of giving definitions of properties. A group G
satisfies property
(1) FC provided every element has only finitely many conjugates.
(2) S1 provided, whenever A and B are finitely generated commensu-
rable subgroups, A∩B has finite index in ⟨NA(A ∩B), NB(A ∩B)⟩.
Here N indicates normalizer.
(3) S2 provided, whenever A and B are finitely generated commen-
surable subgroups, (A ∩B)⟨A,B⟩ has finite index in ⟨A,B⟩. Here
(A ∩B)⟨A,B⟩ is the normal closure of (A ∩B) in ⟨A,B⟩.
(4) �0 provided the set of torsion elements forms a subgroup �(G).
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.68 Nilpotent groups
(5) � provided the set of torsion elements forms a locally finite subgroup
�(G).
(6) U provided roots, when they exist, are unique.
Clearly property S implies each of properties S1 and S2 and property
� implies property �0. Moreover, every property U group is torsion free
and every torsion free group satisfies property � .
Theorem 1.7 (B.H. Neumann [3]). Let G be an FC-group. Then G
satisfies property �0. Moreover, G/�(G) is abelian and torsion free.
Corollary 1. A torsion free FC-group is abelian.
Note that Example 1.3 violates property U, Example 1.4 violates
property � and Example 1.5 violates property �0.
Theorem 1.8. Property S implies property � .
Theorem 1.9. Suppose G is torsion free and satisfies either property S1
or property S2. Then G satisfies property U.
Proof of Theorem 1.8. To show that �(G) forms a subgroup it will suffice
to show that it is closed. Let (a, b) ∈ �(G)2. Let A = ⟨a⟩ and B = ⟨b⟩.
Then A∩B has finite index in each of the finite groups A and B.Hence A∩
B has finite index in ⟨A,B⟩.Thus, ∣⟨A,B⟩∣ = [⟨A,B⟩ : A ∩B] ∣A ∩B∣ <
∞. Since ab ∈ ⟨A,B⟩ it has finite order. Thus, �(G) is closed and forms
a subgroup. We use induction on n to show that, if (g1, ..., gn) ∈ �(G)n
then ⟨g1, ..., gn⟩ is finite. Clearly the result holds for n = 1. Now assume
n > 1 and the result holds for n − 1. Then each of A = ⟨g1, ..., gn−1⟩
and B = ⟨gn⟩ is finite so A ∩ B has finite index in both A and B.
Then A ∩ B has finite index in ⟨A,B⟩ = ⟨g1, ..., gn⟩. Hence, ∣⟨A,B⟩∣ =
[⟨A,B⟩ : A ∩B] ∣A ∩B∣ < ∞. Thus, by induction, we are finished.
Remark 1.10. Essentially the same argument shows that each of the
properties S1 and S2 implies property �0.
Proof of Theorem 1.9. If an = 1 = bn then a = 1 = b since G is torsion
free. So suppose n > 1 and an = bn ∕= 1. We may assume G = ⟨a, b⟩. Let
A = ⟨a⟩ and B = ⟨b⟩. Then A ∩ B is central in G since every element
of A ∩ B commutes with both a and b. Thus, NA (A ∩B) = A, NB(
A ∩ B) = B, and (A ∩B)⟨A,B⟩ = A ∩ B. So, if either property S1 or
S2 is satisfied, then A ∩ B has finite index in G. Now let g ∈ G be
arbitrary. Since A ∩ B is central in G, we must have A ∩ B ≤ NG(g).
This forces [G : NG(g)] < ∞. Since g was arbitrary, G is an FC-group.
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.A. M. Gaglione, S. Lipschutz, D. Spellman 69
It now follows from Corollary 1 that G is abelian. But in the torsion
free abelian group G, an = bn ⇒
(
ab−1
)n
= 1 ⇒ a = b.
2. Some standard and some coined universal algebraic ter-
minology applied to groups
In this section we shall strive to be somewhat more precise about our ter-
minology. We shall reserve the word class to mean any nonempty class
of groups closed under isomorphism. We shall reserve the word property
to mean any property consistent with the group axioms and preserved
under group isomorphism. We shall find it convenient to commit the
abuse of identifying classes and properties both in our verbiage and in
our notation. By a direct product, or a direct power or a subdirect product
we shall mean, in the case of infinitely many factors, the unrestricted
version. That is, we do not insist that at most finitely many coordinates
of an element must be distinct from the identity. In the case of finitely
many factors we shall speak of finite direct products, etc. In what fol-
lows all terminology and notation that is more or less standard will be
assumed familiar to the reader. None the less we have included an ap-
pendix containing a glossary of terms which the reader should feel free to
consult at any point.
By a class operator we mean a function which accepts as inputs arbi-
trary classes and whose image on a class is again a class. We shall adopt
boldface for class operators. A class operator F is a closure operator
provided, for arbitrary classes X and Y,
(1) X ⊆ FX
(2) X ⊆ Y ⇒ FX ⊆ FY
(3) F(FX ) = FX .
The class operators below are (with the sole exception of our usage
of H where he uses Q) as defined in [4]. With the possible exception of
L they are all closure operators.
∙ HX is the class of all homomorphic images of groups in X.
∙ SX is the class of all subgroups of groups in X .
∙ PX is the class of all groups isomorphic to a direct product of a
family of groups in X .
∙ RX is the class of all groups isomorphic to a subdirect product of
a family of groups in X .
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.70 Nilpotent groups
∙ LX is the class of all groups isomorphic to a direct union of groups
in X .
A class is hereditary if it is closed under taking subgroups. Let us
say that a class is fg-hereditary provided it is closed under taking finitely
generated subgroups. We have the rather obvious
Lemma 2.1. If X is fg-hereditary then LX is the class of all groups
whose finitely generated subgroups lie in X .
We then immediately have that L behaves like a closure operator
when restricted to fg-hereditary classes. That is
Corollary 2. Let X and Y be arbitrary fg-hereditary classes. Then
(1) X ⊆ LX
(2) X ⊆ Y ⇒ LX ⊆ LY
(3) L(LX ) = LX .
With the possible exceptions of R0(G,X) andR0(G) (to be defined
in the next section) all of the properties considered in this paper will be
fg-hereditary. A group in LX is said to be locally X while a group in
RX is said to be residually X . Clearly G residually X is equivalent to
the following. For every g ∈ G∖{1} there is a group Hg ∈ X and an
epimorphism 'g : G → Hg such that 'g(g) ∕= 1. In the case when X is
hereditary the insistence that the maps 'g be surjective may be safely
relaxed.
A class V closed under H, S and P is a variety of groups. The
isomorphism class ℰ of the one element group is the trivial variety. All
other varieties V ∕= ℰ are nontrivial. A classical result of Garrett Birkhoff
[Bi] asserts that a class V is a variety if and only if it is the model class
of a set of laws. The variety A of abelian groups is the model class of the
law [x, y] = 1 where [x, y] is the commutator x−1y−1xy. If c ≥ 0 is an
integer, then the variety Nc of groups nilpotent of class at most c is the
model class of the higher commutator law [x1, x2, . . . , xc, xc+1] = 1 where
[x1, . . . , xn] is defined inductively by [x1] = x1, [x1, x2] = x−1
1 x−1
2 x1x2
and [x1, . . . , xn] = [[x1, . . . , xn−1], xn] if n ≥ 3. Note that we identify N0
with ℰ .
If U and V are varieties then their product UV is the class of all
extensions G of a group K ∈ U by a group H ∈ V . The product of
two varieties is again a variety. Multiplication of varieties is associative.
In particular powers of varieties are well-defined. Thus, if d ≥ 0 is an
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.A. M. Gaglione, S. Lipschutz, D. Spellman 71
integer, then Ad is the variety of all groups solvable of length at most d
and where, by convention, A0= ℰ and A1= A.
For each positive integer n the Burnside variety ℬn is the variety
determined by the law xn = 1. The union N of the chain N0 ⊆ N1 ⊆
N2 ⊆ ⋅ ⋅ ⋅ is the class of nilpotent groups and the union S of the chain
A0 ⊆ A1 ⊆ A2 ⊆ ⋅ ⋅ ⋅ is the class of solvable groups. Here is a good point
to advertise
Theorem 2.2 (Fine, Gaglione and Spellman [5]). The class X is a union
of varieties if and only if it is closed under taking subgroups, homomorphic
images and direct powers; furthermore, X is a direct union of varieties if
and only if additionally it is closed under taking finite direct products.
Every nontrivial variety admits free objects of all ranks. If V is any
nontrivial variety and r is any cardinal then Fr(V) shall denote a fixed
but arbitrary group free of rank r in V . For fixed r and V ∕= ℰ this is
unique up to isomorphism.
We now coin some terminology for our purposes.
Definition 2.3. Let X be a property. A G group satisfies
strong X or (HS)−1X
provided whenever G0 is a subgroup in G and G∗
o is a homomorphic image
of G0 it is the case that G∗
0 satisfies X . X is a strong property provided
(HS)−1X = X .
Thus property S, which is not preserved in homomorphic images, is
not strong. On the other hand, in view of Theorem 2.2, each of nilpotence
and solvability is strong. (Admittedly the strength of nilpotence and
solvability is rather obvious without explicit appeal to Theorem 2.2.) We
make the transparent observations
(1) strong X ⇒ X and
(2) If X ⇒ Y, then strong X =⇒ strong Y.
3. Nilpotent groups and more
Our reference for nilpotent groups shall be [1]. It is well-known that
nilpotent groups satisfy property � and that torsion free nilpotent groups
satisfy property U. Probably less well-known is the fact, proven in [2],
that nilpotent groups satisfy property S (from which the aforementioned
properties � and U, in the torsion free case, follow). This is a consequence
of a technical result of Mal’cev.
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.72 Nilpotent groups
Lemma 3.1 (Mal’cev [1]). Let X be a finite set of generators for a nilpo-
tent group G and let H be a subgroup in G. Then H has finite index in
G if and only if for every x ∈ X there is a positive integer n(x) such that
xn(x) ∈ H.
Corollary 3. Nilpotent groups satisfy property S.
Proof. Suppose A and B are finitely generated commensurable subgroups
of a nilpotent group G. Say A = ⟨a1, ..., ap⟩ and B = ⟨b1, ..., bq⟩. Suppose
ami ∈ A ∩ B, i = 1, ..., p and b
nj
j ∈ A ∩ B, j = 1, ..., q where the mi and
nj are positive integers. Since {a1, ..., ap, b1, ..., bq} generates ⟨A,B⟩ we
conclude from Lemma 3.1 that [⟨A,B⟩ : A ∩B] < ∞.
Note that nilpotent groups actually satisfy strong property S since
nilpotence is strong. Now following the development (but not the nota-
tion) in [2] let us say that a group with a finite generating set X satisfies
R0(G,X) provided whenever H is a subgroup in G it is the case that H
has finite index in G if and only if for each x ∈ X there is a positive
integer n(x) such that xn(x) ∈ H. If G is any finitely generated group we
say that G satisfies R0(G) provided it satisfies R0(G,X) for every finite
set X of generators. Finally, G satisfies R0 provided it is finitely gener-
ated and every finitely generated subgroup H ≤ G satisfies R0(H). By
its very definition property R0 is fg-hereditary. Property R was defined
in [2] to be our property R0 and in [7] to be our property LR0. Note
that these formulations coincide on the class of finitely generated groups.
Thus, we choose the latter. Explicitly —
Definition 3.2. G satisfies property R provided whenever X is a finite
subset of G and H is a subgroup in ⟨X⟩ it is the case that H has finite
index in ⟨X⟩ if and only if for every x ∈ X there is a positive integer
n(x) such that xn(x) ∈ H.
Examples of non-nilpotent property R groups were given in [2] and
[7]. Clearly the proof of Corollary 3 yields
Theorem 3.3. Property R implies property S.
In fact, since it was proven in [7] that property R is preserved in ho-
momorphic images, property R is strong and hence property R implies
strong property S. Now since property S implies property � strong prop-
erty S implies strong property � . The chain of implications gives that
property R implies strong property � . In fact it was proven in [7] that
strong property � is equivalent to a weakened version of property R.
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.A. M. Gaglione, S. Lipschutz, D. Spellman 73
Definition 3.4 ([7]). The G group satisfies weak property R provided
whenever X is a finite subset of G and K is a subgroup of ⟨X⟩ which is
normal in ⟨X⟩ it is the case that K has finite index in ⟨X⟩ if and only if
for each x ∈ X there is a positive integer n(x) such that xn(x) ∈ K.
Observe that essentially the proof of Corollary 3 establishes that a
weak property R group satisfies each of the properties S1 and S2. Hence
(from either S1 or S2) it follows that a torsion free weak property R group
is a U-group.
Theorem 3.5 ([7]). Weak property R is equivalent to strong property � .
In [7] the authors posed the question of whether or not weak property
R implies property R. In the next section we present an explicit coun-
terexample establishing that weak property R is strictly weaker than
property R.
4. The counterexample
An easy induction on the solvability length establishes the classic
Lemma 4.1 (Phillip Hall). A finitely generated solvable group of finite
exponent is finite.
Now suppose G is a solvable group whose commutator subgroup G′
has finite exponent e. Suppose g and ℎ are torsion elements in G. Then
certainly there are positive integers m and n such that gm ≡ ℎn ≡
1(mod G′). Let m0 and n0 be the least such positive integers and let
L be the least common multiple of m0 and n0. Then (gℎ)L ≡ gLℎL ≡
1(mod G′). It follows that (gℎ)Le = ((gℎ)L)e = 1 since G′ has exponent
e. Thus, the set �(G) of torsion elements in G forms a subgroup. Now
let (g1, ..., gk) ∈ �(G)k be a finite tuple of torsion elements. Then, for all
i = 1, ..., k there is a positive integer ni such that gni
i ≡ 1(mod G′). Let
mi be the least such positive integer and let L be the least common mul-
tiple of m1, ...,mk. Now suppose is x ∈ ⟨g1, ..., gk⟩ is arbitrary. We may
assume x ≡ g�11 ⋅ ⋅ ⋅ g�kk (mod G′). Then xL ≡ g�1L1 ⋅ ⋅ ⋅ g�kLk ≡ 1(mod G′).
Consequently xLe = 1. It follows that the finitely generated solvable
group ⟨g1, ..., gk⟩ has finite exponent. Hence, by Lemma 4.1, it is finite.
Therefore �(G) is locally finite and hence G satisfies property � . But
it is easy to see that solvable with derived group of finite exponent is a
strong property. Hence, G satisfies strong property � . Equivalently, G
satisfies weak property R. We have proven
Theorem 4.2. A solvable group whose commutator subgroup has finite
exponent satisfies weak property R.
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.74 Nilpotent groups
Now consider the product variety ℬ2A. First we concentrate on the
left hand factor ℬ2. That is the variety of all groups satisfying the law
x2 = 1. But these are precisely the elementary abelian 2-groups. That is,
vector spaces over the two element field. Hence, ℬ2⊆ A and ℬ2A ⊆ A2
so every group in ℬ2A is metabelian. Furthermore, if G ∈ ℬ2A, there is
a short exact sequence
1 → K → G → H → 1
where K ∈ ℬ2 and H is abelian. Then G′ ≤ K must satisfy the law
x2 = 1. Thus, every group in ℬ2A is a metabelian group whose commu-
tator subgroup has exponent dividing 2. In view of Theorem 4.2 every
group ℬ2A satisfies weak property R. We shall prove that the group
G = F2(ℬ2A) free of rank 2 in ℬ2A violates property R.
Consider the free product X ∗ Y of abelian groups X and Y . By
problems 23, 24 and 34 pages 196 and 197 of [8], the commutator subgroup
of X ∗ Y is freely generated by the commutators [x, y] as x and y vary
independently over X∖{1} and Y ∖{1} respectively. Thus, the derived
group F̂ ′ of F̂ = ⟨â; ⟩ ∗
〈
b̂;
〉
=
〈
â, b̂;
〉
is freely generated by the
commutators [âm, b̂n] as (m,n) varies over (ℤ∖{0})2. Now suppose that
{a, b} is a basis relative to ℬ2A for G = F2(ℬ2A). Under the epimorphism
F̂ → G determined by â 7→ a, b̂ 7→ b, F̂ ′ maps onto G′. It follows that,
as (m,n) varies over (ℤ∖{0})2, the commutators [am, bn] form a basis
for the multiplicatively written elementary abelian 2-group G′. Hence,
apart from the order of the factors in G′ every element of G is uniquely
expressible in the form
a�b�
∏
(m,n)∈(ℤ∖{0})2
[am, bn]
(m,n)
where � and � are integers,
(m,n) ∈ {0, 1}and all but finitely many
(m,n) are zero. (See also Corollary 21.13 of [6] and its proof.) Moreover,
since G′ is abelian, the order of the factors in
∏
(m,n)∈(ℤ∖{0})2
[am, bn]
(m,n)
is immaterial. Now, since G is free in ℬ2A on {a, b} the assignment
a 7→ a2, b 7→ b2 extends uniquely to an epimorphism � from G onto the
subgroup
〈
a2, b2
〉
. Applying � to a�b�
∏
(m,n)∈(ℤ∖{0})2
[am, bn]
(m,n) we get
a2�b2�
∏
(m,n)∈(ℤ∖{0})2
[a2m, b2n]
(m,n).
By uniqueness, � has trivial kernel and hence is an isomorphism. It
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.A. M. Gaglione, S. Lipschutz, D. Spellman 75
follows that every element of
〈
a2, b2
〉
is uniquely expressible in the form
a2�b2�
∏
(m,n)∈(ℤ∖{0})2
[a2m, b2n]
(m,n).
More importantly for our purposes, it follows that
a�b�
∏
(m,n)∈(ℤ∖{0})2
[am, bn]
(m,n)
lies in
〈
a2, b2
〉
if and only if both � and � are even and, addition-
ally,
(m,n) = 0 whenever at least one of m or n is odd. Now let
k1 ∕= k2 be integers. Recalling that [x, y]−1 = [x, y] for all commuta-
tors [x, y] ∈ G′ since G′ is an elementary abelian 2-group,we have that
[a, b2k1+1][a, b2k2+1]−1 = [a, b2k1+1][a, b2k2+1] does not lie in H =
〈
a2, b2
〉
.
Thus, H [a, b2k1+1] ∕= H[a, b2k2+1] . Consequently H =
〈
a2, b2
〉
has
infinite index in G. Hence, G violates property R.
5. Questions
We conclude with two questions each of which we conjecture has a neg-
ative answer.
Question 1: Does weak property R imply property S?
Question 2: Does strong property S imply property R?
6. Appendix (Glossary)
1. Direct union of groups
A group G is the direct union of a family ℋ of subgroups provided
(1) G = ∪ ℋ and
(2) For all H0, H1 ∈ ℋ there is H ∈ ℋ such that ⟨H0, H1⟩ ≤ H.
2. Direct union of classes
The class Y is the direct union of the family ℱ of subclasses provided
(1) Y= ∪ ℱ and
(2) For all X0,X1 ∈ ℱ there is X ∈ ℱ such that X0 ∪ X1 ⊆ X .
3. Subdirect product of a family of groups
Let I be a nonempty set and let (Gi)i∈I be a family of (not necessarily
distinct) groups indexed by I. Let P =
∏
i∈I
Gi be the direct product
of the indexed family. That is, P consists of all choice functions g :
I →
∪
i∈I
Gi, i 7→ gi ∈ Gi for all i ∈ I with group operations defined
componentwise. For each fixed i ∈ I let be pi : P → Gi projection onto
the i-th coordinate, pi(g) = gi. A subgroup G ≤ P is a subdirect product
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.76 Nilpotent groups
of the family (Gi)i∈I provided, for all i ∈ I, the restriction pi ∣Gmaps G
onto Gi.
4. Relatively free groups and verbal subgroups
Let V be a nontrivial variety of groups. Suppose X is a set of gen-
erators for a group Φ in V . Then Φ is free in V on X provided every
assignment of values X → G from X into a group G in V extends (nec-
essarily uniquely) to a homomorphism Φ → G. It follows from abstract
nonsense that, if r is any cardinal, then Fr(V) is unique up to isomor-
phism whenever it exists. Existence is deduced by applying the concept
of verbal subgroup defined below.
Let G be any group – not necessarily in V . Let V (G) be the family
of subgroups K normal in G such that G/K lies in V . Clearly G ∈
V (G) so V (G) is nonempty. Now we get a homomorphism defined by
' : G →
∏
K∈V (G)
(G/K) defined by '(g)(K) = Kg. Since V is a variety it
is closed under direct products and subgroups so the image '[G] lies in
V . Hence, Ker(') lies in V (G) . But Ker(') = ∩V (G) is then the unique
minimum element of V (G) . By definition ∩V (G) is the verbal subgroup
V (G) corresponding to the variety V .
In practice, if w (x) = 1 is a set of laws (on tuples x of variables)
determining V , then V (G) is the subgroup of generated by the elements
w(g) as w (x) = 1 varies over the laws and g varies over tuples from G.
If r is a cardinal and F is absolutely free (i.e. free in the variety of all
groups) on {a�+1 : 0 ≤ � < r}, then one can prove that F/V (F ) is free in
V on {V (F )a�+1 : 0 ≤ � < r}.
Remark 6.1. One can show that V = ℬ2A is determined by the law
([x, y][z, w])2 = 1.
Thus, F2(ℬ2A) is F̂ /V (F̂ ) where F̂ =
〈
â, b̂;
〉
and V is the verbal sub-
group operator corresponding to the law ([x, y][z, w])2 = 1.
References
[1] G. Baumslag, Lecture Notes on Nilpotent Groups, AMS, Providence, 1971.
[2] G. Baumslag, O. Bogopulski, B. Fine, A.M. Gaglione, G. Rosenberger and D.
Spellman, “On some finiteness properties in infinite groups,” Algebra Colloquium,
15 (2008), 1-22.
[3] B.H. Neumann, “Groups with finite classes of conjugate elements,” Proc. London
Math. Soc. 3 (1), (1961), 178 –187. [Bi] G. Birkhoff, “On the structure of abstract
algebras,” Proc. Cambridge Philos. Soc. 31, (1935), 433 – 454.
[4] P.M. Cohn, Universal Algebra, Harper and Row, New York, 1965.
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.A. M. Gaglione, S. Lipschutz, D. Spellman 77
[5] B. Fine, A.M. Gaglione and D. Spellman, “Unions of varieties and quasivarieties,”
Combinatorial Group Theory, Discrete Groups and Number Theory, AMS, Con-
temporary Mathematics Series (421), 2006, 113 – 118.
[6] H. Neumann, Varieties of Groups, Springer – Verlag, Berlin, 1967.
[7] S. Lipschutz and D. Spellman, “An application of a group of Ol’sanskii to a
question of Fine et al,” World Scientific, 2009, B. Fine, G. Rosenberger and D.
Spellman editors, 201-211.
[8] W. Magnus, A. Karrass and D. Solitar, Combinatorial Group Theory, John Wiley
and Sons, New York, 1966.
Contact information
A. M. Gaglione Department of Mathematics
U.S. Naval Academy
Annapolis, MD 21402 USA
E-Mail: amg@usna.edu
S. Lipschutz Department of Mathematics
Temple University
Philadephia, PA 19122 USA
E-Mail: seymour@temple.edu
D. Spellman Department of Mathematics
Temple University
Philadephia, PA 19122 USA
E-Mail: spellman@temple.edu
Received by the editors: 23.05.2009
and in final form 23.05.2009.
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