Algebra and Discrete Mathematics RESEARCH ARTICLE
Number 3. (2006). pp. 55 – 70
c© Journal “Algebra and Discrete Mathematics”
Absolutely ubiquitous groups
Aleksander Ivanov and Krzysztof Majcher
Communicated by V. I. Sushchansky
Abstract. We characterise absolutely ubiquitous groups in
the class of model complete ω-categorical groups.
1. Introduction
Let M be a countable structure over a finite language. The age of M , de-
noted by J (M), is the set of all isomorphism types of finite substructures
ofM . The structureM is absolutely ubiquitous if it is uniformly locally fi-
nite and for any countable locally finite structure N with J (M) = J (N)
we have M ∼= N . This notion has been introduced by P. Cameron (see
[3]). It covers the case of ω-categorical universally axiomatisable struc-
tures already studied by E. A. Palyutin in [8].
In fact in [8] model-theoretic investigations of absolutely ubiquitous
structures have been initiated. It turns out that some basic problems
studied there are still open. It is still unknown if absolutely ubiquitous
stuctures are ω-stable (or even stable). On the other hand H. D. Macpher-
son has noticed in [7] that absolutely ubiquitious structures are ω-catego-
rical, model complete and do not have the strict order property. Together
with I. Hodkinson he has described all absolutely ubiquitous structures
in relational languages. Recently Gabor Sagi has extended this (and [5])
to structures with linear growth of substructures [9].
In [7] Dugald Macpherson has begun investigations of absolutely ubiq-
uitous groups. He has shown that such a group has a characteristic
The research was supported by KBN grant 1 P03A 025 28.
2000 Mathematics Subject Classification: 20A05, 20F18, 20K30, 03C45.
Key words and phrases: absolutely ubiquitous structures, nilpotent groups,
ω-categorical modules.
56 Absolutely ubiquitous groups
subgroup of finite index which is a direct product of elementary abelian
groups of infinite rank. Using this it can be shown that an absolutely
ubiquitous group is ω-stable.
In our paper we continue this theme in order to obtain an algebraic
characterisation of absolutely ubiquitous groups. This question was also
discussed in [7] and some partial results were given there.
The idea of our method is that we study absolutely ubiquitous groups
in the language extended by automorphisms L′ = (·, α1, ..., αn) and we
show that some stronger version of the theorem of Macpherson cited
above holds in this situation: an absolutely ubiquitous (G,α1, ..., αn) has
a characteristic abelian subgroup A < G of finite index with the following
properties. Let γ1, ..., γm be the automorphisms of A which are the con-
jugations induced by a transversal of G/A. Then (A,α1, ..., αn, γ1, ..., γm)
is absolutely ubiquitous. As a module over R = Z[α1, ..., αn, γ1, ..., γm]
the group A is a direct sum of finitely many R-modules of the form Bω,
where B is irreducible, i.e. does not have non-trivial submodules (this
follows just from absolute ubiquity of A). Then we apply these results to
some characterisation of absolutely ubiquitous groups in terms of their
submodules of finite index.
We now give some basic definitions and facts. We fix a language L
and study absolutely ubiquitous L-structures. Note that if M0,M1,M2
are L-structures where M0 is absolutely ubiquitous, M2 |= Th(M0) and
M0 ⊆ M1 ⊆ M2, then J (M0) = J (M1) = J (M2) and M1 |= Th(M0)
(see [7]).
We do not use any involved model-theoretical material. In particular
we do not assume that the reader knows what a stable (or ω-stable)
theory is. On the other hand to follow the logical structure of the paper
it is worth knowing that ω-stable theories are stable and stable structures
do not have the strict order property. The latter is defined as follows:
there is a natural number k and a 2k-ary formula φ(x¯, y¯)
(possibly with parameters) which defines a partial order on
Mk with an infinite chain.
In fact our model-theoretic arguments are restricted to some applications
of the compactness theorem and some basic properties of ω-categorical
and model complete theories. We remind the reader that a theory T is
model complete if and only if any formula φ of the language L is equivalent
with respect to T to some existential formula ψ and to some universal
formula θ. When φ(x¯, y¯) is a formula, M is a structure and b¯ is a tuple
from M , then we denote by φ(M, b¯) the set {a¯ ∈M : M |= φ(a¯, b¯)}.
We now give some preliminaries concerning ω-categorical groups. In
this paper we study groups (G, ·, α1, ..., αm) in the language extended
A. Ivanov, K. Majcher 57
by automorphisms α1, ..., αm. By α¯ we denote α1, ..., αm. Our basic
algebraic notation is standard (the same as in [2] and [7]). For example
[x, y] is an abbreviation for x−1y−1xy.
Since all Aut(G)-invariant relations of an ω-categorical group are de-
finable (i.e. of the form φ(M, b¯)), all characteristic subgroups of G are
definable. It is also worth noting that for any natural k such a group
has only finitely many Aut(G)-invariant k-ary relations. We will often
use Lemma 5.1 from [7] that, if F is a finite subgroup of an ω-categorical
nilpotent group H, then the centralizer CH(F ) is infinite. The follow-
ing theorem of Macpherson from [7] will play an important role in our
arguments.
Theorem A. If G is an ω-categorical group and Th(G) does
not have the strict order property, then G has a characteristic
nilpotent subgroup of finite index.
This theorem generalises one of the main results of [2].
Let (G, p1, ..., pm, P1, ..., Pk) be a group G in the language extended
by functions pi and relations Pj ⊆ Glj . For A,B ⊂ G we denote by
〈A〉pi1 ,...,pij ,B the smallest subgroup of G, containing A and closed under
the functions pi1 , ..., pij 1 and conjugation by elements of B. Generalising
Lemma 3.2 from [7] we obtain the following lemma.
Lemma 1.1. If (G, ·, p1, ..., pm, P1, ..., Pk) is an ω-categorical structure,
then there is a formula ψ(x, y) such that (G, p¯, P¯ ) |= ψ(g, h) if and only
if h ∈ 〈g〉p¯,G.
Proof. Any element of 〈g〉p¯,G can be expressed by a term built from ·,
all pi and xy (with y ∈ G). By ω-categoricity the depth of such terms
can be bounded by a natural number. This allows us to find ψ as in the
formulation.
2. Submodules of finite index
In this section we develop Theorem 1.3 from [7] by some inspection of
Macpherson’s proof.
Theorem 2.1. Let (G, α¯, P1, ..., Pk) be an absolutely ubiquitous group in
a language extended by automorphisms and relations. Then G has a char-
acteristic subgroup of finite index which is a direct product of elementary
abelian groups of infinite rank.
1p¯-closed
58 Absolutely ubiquitous groups
The situation with this theorem is slightly unusual. The point is that
there is no reason why a reduct of an absolutely ubiquitous structure is
absolutely ubiquitous. Thus the original Macpherson’s theorem (proved
for the pure group language) cannot be applied here. We have found that
the proof given in [7] works in our case with obvious changes: basically
we add everywhere the condition of α¯-closedness of subgroups. On the
other hand the proof from [7] is quite long and involved, and we are
not sure that the reader can easily trust this recipe of the corresponding
conversion. Thus we have decided to include our version of Macpherson’s
proof into the paper.
Our starting point is the same as in [7]. This is the observation
that if (G, α¯, P1, ..., Pk) is absolutely ubiquitous then G has a nilpotent
characteristic subgroup N of finite index. This follows from Theorem A
and the fact that G is ω-categorical and does not have the strict order
property. Now let N be a nilpotent characteristic subgroup of G of finite
index. We want to prove that N has an abelian characteristic subgroup
of finite index. The key result is as follows.
Lemma 2.2 (Main Lemma). Let M be a characteristic subgroup of N
such that N/M is of nilpotency class two. Then N/M has an abelian
characteristic subgroup of finite index.
Proof. By inspection of the proof of Theorem 1.3 from [7]. The
presentation below is divided into Lemmas 2.3 - 2.7 and the concluding
part of the proof.
Let Θ = {t1, .., tp} be a transversal of G/N . We shall suppose for a
contradiction that |N/M : Z(N/M)| is infinite. By ω-categoricity there
are formulas φM (x), φN (x) and φZ(x) in the language of groups, such
the following holds:
G |= φM (x) if and only if x ∈M
G |= φN (x) if and only if x ∈ N
G |= φZ(x) if and only if xM ∈ Z(N/M).
Let Ω be the set of all non-trivial term functions of the language 〈α1, ..., αm〉
in G. By ω-categoricity Ω is finite. Modifying the corresponding place
from [7] we say that g ∈ N is good if
(i) 〈g〉α¯,GM/M is finite ;
(ii) there is H ≤ G with M ≤ H such that g 6∈ H, G =
〈g〉α¯,GH, the groups H and G are isomorphic in the lan-
guage 〈·, α1, ..., αm, P1, ..., Pk〉, and G |= φM ([(γ(g))b, h]) for
all h ∈ H ∩N , b ∈ G and γ ∈ Ω.
A. Ivanov, K. Majcher 59
In this case we say that the pair (H, g) is good in G. Since the set of good
elements is invariant under Aut(G, α¯, P¯ ), there is a formula ψgd(x) such
that (G, α¯, P¯ ) |= ψgd(h) if and only if h is good. This formula enables us
to talk about good elements in isomorphic copies of (G, α¯, P¯ ).
Lemma 2.3. There are ω-sequences (Gi : i < ω) of α¯-closed subgroups
of G containing M and (gi : i < ω) of good elements of G such that
(i) G0 = G and Gi+1 ≤ Gi for all i < ω
(ii) ⋂(Gi : i < ω) contains no good elements of G
(iii) for any i < ω the pair (Gi+1, gi) is good in Gi
(iv) for any i < ω, gi has infinitely many translates under
Aut(G, α¯, P1, ..., Pk) among (gj : j < ω).
Proof. We show first that there is a pair (H, g) which is good in
(G, α¯, P¯ ). Extend the language L to the language L∗ = L∪{b, ai : i < ω},
where the ai are interpreted bijectively by the elements of G. Let T ∗
denote the complete theory of (G, α¯, P¯ ) over L∗ \ {b}. For each i < ω,
let σi be the sentence
∧
{¬φM ((γ(b))ta−1i ) ∧ φN ((γ(b))t) ∧ [φN (ai) →
→ φM ([(γ(b))t, ai])] : γ ∈ Ω, t ∈ Θ}.
Claim 1. T ∗ ∪ {σi : i < ω} is consistent.
It suffices to show that for any n < ω, T ∗ ∪ {σi : i ≤ n} is con-
sistent. Let a1, ..., an ∈ G and {a1, ..., an} ∩ N = {a1, ..., al}. Then
F1 = 〈a1, ..., an〉Ω,ΘM/M and F2 = 〈a1, ..., al〉Ω,ΘM/M are finite. From
Lemma 5.1 of [7] 2 CN/M (F2) is infinite. Hence, for each n < ω, T ∗∪{σi :
i ≤ n} has an interpretation for b from CN/M (F2) \ F1.
By Claim 1 T ∗∪{σi : i < ω} has a countable model G˜, which contains
G. Let G∗ = 〈G, b〉α¯. Then 〈b〉α¯,G is finite and (G, α¯, P¯ ) is an elementary
substructure of (G∗, α¯, P¯ ).
Claim 2. The pair (G · φM (G∗), b) is good in G∗.
To see condition (i) of the definition present an element g of G∗ in
the form g = g′b′tg′′, where g′ ∈ φN (G), g′′ ∈ φM (G∗), t ∈ 〈Θ〉 and
b′ ∈ 〈b〉Ω,G. Then we have modulo φM (G∗): bg = t−1b′−1bb′t ∈ 〈t1, ..., tp〉·
〈b〉Ω,G and the latter subgroup is finite. Thus 〈b〉Ω,G∗φM (G∗)/φM (G∗) is
finite.
To see condition (ii) note that G ≤ G · φM (G∗) ≤ G∗ ≤ G˜, G |= T ,
G˜ |= T and G∗ ∼= G · φM (G∗). The rest is easy.
2see also Introduction
60 Absolutely ubiquitous groups
We now repeat the arguments from the corresponding part of the
proof of Lemma 5.3 from [7]. By Claim 2 there is a pair (H, g) which is
good in G. Since H ∼= G, we may apply this remark to H and to the
corresponding subgroups of H. It follows that G contains infinitely many
good elements; let these be listed as {hi : i < ω}. Let O1, ..., Ol be the
set orbits of Aut(G, α¯, P¯ ) on G which intersect {hi : i < ω} non-trivially.
Clearly each such intersection is infinite.
Suppose that after r steps we have constructed a finite chain Gr <
Gr−1 < ... < G0 = G together with elements gr−1, ..., g0 satisfying the
conditions of the lemma. Since (Gr, α¯, P¯ ) is an elementary submodel
of (G, α¯, P¯ ), the group Gr contains good elements and these elements
are precisely the good elements of G which lie in Gr. Let r ≡ s mod l
where s ∈ {1, ..., l}. Choose the least gr among the elements of the set
{hi : i < ω} which lie in Gr
⋂Os, and choose Gr+1 ≤ Gr such that the
pair (Gr+1, gr) is good in Gr. This construction yields the lemma.
For the set of gi from Lemma 2.3 put V = 〈〈gi〉Ω,G : i < ω〉M .
Lemma 2.4. The group V/M is abelian, and lies in Z(N/M).
Proof. As in the proof of the previous lemma any g ∈ G is of the
form g′0g′1...g′ig′t, where g′ ∈ φN (Gi+1), g′j ∈ 〈gj〉Ω,Gj for j = 1, ..., i and
t is an element of a transversal of Gi+1/φN (Gn+1) (it is clear that if
g ∈ φN (G) then t = 1). Thus 〈gi〉Ω,GM/M = 〈gi〉Ω,GiM/M and V/M is
a commuting product of finite groups 〈gi〉Ω,GM/M . If we can show that
each group 〈gi〉Ω,GM/M is abelian, we see 〈gi〉Ω,GM/M ≤ Z(N/M).
Suppose that 〈gi0〉Ω,GM/M is non-abelian, and let ρ(x) be a formula
such that G |= ρ(g) iff g is good and 〈g〉Ω,GM/M is non-abelian. By the
choice of gi, the formula ρ(x) is satisfied by infinitely many elements, say
{gik : ik < ω} from {gi : i < ω}. Write hk := gik for all k < ω.
Claim 1. Let B be the least 0-definable subgroup of G with M ≤ B ≤ N
such that whenever g is good in G, γ ∈ Ω and h ∈ G we have [g, (γ(g))h] ∈
B. Then |B : M | is finite.
The proof of this claim is a slight modification of the proof of Claim
1 from Lemma 5.4 from [7]. Assuming the contrary we find γ ∈ Ω and an
infinite sequence of elements of N/M of the form [hi, γ(hkii )]M , ki ∈ G.
Assuming for simplicity that this sequence depends on all i ∈ ω consider
the elements h0M,h0h1M,h0h1h2M, .... Conjugating them by products
of elements γ(hkii )M we see that they lie in finite conjugacy classes of
arbitrary large size (contradicting ω-categoricity).
Claim 2. Put h = h0...hk for k ∈ ω. Then G |= ρ(h).
It suffices to repeat the corresponding part of [7]. Find g ∈ G and
γ ∈ Ω such that [(γ(h0))g, h0]M 6= M . Then [(γ(h0))g, h]M 6= M , so
hM 6∈ Z(N/M). Assume hk = gr. Then N/M = ((Gr+1 ∩ N)/M) ·
A. Ivanov, K. Majcher 61
〈〈gj〉Ω,GM/M : 0 ≤ j ≤ r〉 and h 6∈ Gr+1. Since the pair (h,Gr+1) is
good in 〈h,Gr+1〉Ω, the element h is good in G. If 〈h〉Ω,GM/M were
abelian, then by arguments above, h would satisfy φZ(x) in 〈h,Gr+1〉Ω.
Since the latter is an elementary submodel of G, h would satisfy φZ(x)
in G. This is a contradiction.
As in [7] by an application of Ramsey’s Theorem we may suppose that
all pairs (hiM,hjM) for i < j < ω, lie in the same orbit of Aut(G, α¯, P¯ ).
This assumption and the finiteness of G/N and B/M imply that
for any g ∈ G and γ ∈ Ω there is bgγ ∈ B such that [(γ(hi))g, hi] ∈
bgγM for all i < ω.
Now let s = |B : M |. For any g ∈ G, and γ ∈ Ω
[(γ(h0...hs−1))g, (h0...hs−1)]M =
= Πsi=0[(γ(hi))g, hi]M = bsgγM = (bgγM)s = M.
This contradicts Claim 2.
Now put Qˆ = ⋂(Gi/M : i < ω) and Q =
⋂(Gi : i < ω). Clearly Q
is Ω-closed and contains no good elements of G. We want to show that
every finite subgroup of N/M has finite normal closure in N/M . The
argument of Lemma 5.5 [7] shows, that
Lemma 2.5. [N/M,N/M ] ≤ Qˆ.
Proof. Any two h and h′ ∈ N can be presented as h = hizi and
h′ = h′iz′i with hi, h′i ∈ Gi and zi, z′i ∈ 〈g0, ..., gi−1〉Ω,G. Then by Lemma
2.4 [h, h′]M = [hi, h′i]M ∈ Gi/M . Since i was arbitrary, [h, h′]M ∈ Qˆ.
Lemma 2.6. Qˆ ∩ Z(N/M) is finite.
Proof. Suppose Qˆ ∩ Z(N/M) is infinite. Note that if g ∈ Q then
G |= ¬ψgd(g). To obtain a contradiction we want to show that G contains
a good element b with G |= ¬ψgd(b). Find a formula ψ(x) such that
G |= ψ(g) if and only if 〈g〉Ω,GφM (G)/φM (G) is abelian. Extend the
language L to the language L∗ = L ∪ {b, ai : i < ω}, where the ai are
interpreted bijectively by the elements of G. Let T ∗ denote the complete
theory of (G, α¯, P¯ ) over L∗ \ {b}. For each i < ω, let σi be the sentence
∧
{¬φM ((γ(b))ta−1i ) ∧ φN ((γ(b))t) ∧ ¬ψgd((γ(b))t) ∧ ψ(b)∧
[φN (ai) → φM ([(γ(b))t, ai])] : γ ∈ Ω, t ∈ Θ}.
To see that T ∗ ∪ {σi : i < ω} is consistent let us show that for any
n < ω, T ∗ ∪ {σi : i ≤ n} is consistent. Let a1, ..., an ∈ G. Then
62 Absolutely ubiquitous groups
F = 〈a1, ..., an〉Ω,ΘM/M is finite. Hence, for each n < ω, T ∗ ∪ {σi :
i ≤ n} has an interpretation for b from (Qˆ ∩ Z(N/M))M \ FM (to see
〈b〉Ω,GM/M ⊆ Z(N/M) for such an interpretation, apply the fact that
bM ∈ Z(N/M)).
Find a countable model G˜ |= T ∗ ∪{σi : i < ω} which contains G. Let
G∗ = 〈G, b〉Ω. As in the proof of Lemma 2.3 the pair (G · φM (G∗), b) is
good in G∗ and G∗ |= ¬ψgd(b), a contradiction.
Lemma 2.7. If F is a finite subgroup of N/M , then the group F · (Qˆ ∩
Z(N/M)) is a finite normal subgroup of N/M .
Proof of this statement is the same as in [7]. The finiteness follows
from Lemma 2.6. By Lemma 2.5 for any f ∈ F and any h ∈ N/M
we have h−1fhf−1 ∈ Qˆ. By the assumption that N/M has nilpotency
class two, h−1fhf−1 ∈ Z(N/M). Thus there is z ∈ Qˆ ∩ Z(N/M) with
h−1fh = zf .
Proof of Lemma 2.2. We shall show that G contains a good element
b with bM 6∈ Z(N/M). This will contradict Lemma 2.4. Extend the
language L to the language L∗ = L ∪ {b, ai : i < ω}, where the ai are
interpreted bijectively by the elements of G. Let T ∗ denote the complete
theory of (G, α¯, P¯ ) over L∗ \ {b}. For each i < ω, let σi be the sentence
∧
{¬φM ((γ(b))ta−1i ) ∧ φN ((γ(b))t)∧
[φN (ai) → φM ([(γ(b))t, ai])] ∧ ¬φZ(b) : γ ∈ Ω, t ∈ Θ}.
It suffices to show that for any n < ω, T ∗ ∪ {σi : i ≤ n} is consis-
tent. Let a1, .., an ∈ G, where {a1, .., an} ∩ N = {a1, ..., al}. Then
F1 = 〈a1, ..., an〉Ω,ΘM/M and F2 = 〈a1, ..., al〉Ω,ΘM/M are finite. By
Lemma 2.7, CN/M (F2 · (Qˆ ∩ Z(N/M))) has finite index in N/M and by
asumptions Z(N/M) has infinite index in N/M . Thus we can find an
interpretation for b in CN/M (F2 ·(Qˆ∩Z(N/M)))M \(Z(N/M)M∪F1M).
The rest is as above.
Proof of Theorem 2.1. It follows from Lemma 2.2 that if G has a
nilpotent characteristic subgroup of finite index and nilpotency class l,
then G also has a characteristic subgroup of finite index and nilpotency
class l − 1. By induction we obtain that if (G, α¯, P¯ ) is an absolutely
ubiquitous group, then G has a characteristic abelian subgroup A of finite
index. Let Θ be a set of representatives of G/A. Let pi (1 ≤ i ≤ q) be
the primes dividing the orders of elements of A, and let S1, ..., Sq be
the corresponding Sylow pi-subgroup of A; then A = S1 × ... × Sq. For
i = 1, ..., q let Vi be the subgroup of Si consisting of elements of order pi.
To complete the proof of the theorem we must show that for any i, |Si/Vi|
A. Ivanov, K. Majcher 63
is finite. If |Si/Vi| is infinite, put p := pi, S := Si and V := Vi. Let φ(x)
be the formula x ∈ A ∧ xp = 1 ∧ x 6= 1 ∧ (∃y ∈ A)(yp = x). Extend L to
the language L∗ ∪ {b, ci : i < ω}, where ci are interpreted bijectively by
the elements of G. Let T ∗ be the theory of (G, α¯, P¯ ) over L∗ \ {b}. Since
S/V is infinite, for any finite subset of Σ∗ = T ∗ ∪ {b 6= ci} ∪ {φ(b)}, an
interpretation for b can be found in V .
Let G∗ be a model of Σ∗ containing G, and put G˜ = 〈G, b〉α¯. Then
G˜ |= φ(b).
On the other hand the group 〈A, b〉α¯,Θ is of index |G : A| in G˜; thus
is defined by the formula for A in G. Now it is easily seen that this
formula cannot be realised by an element a in G˜ with ar = b. This is a
contradiction.
Remark. Note that the relations Pi from the formulation do not
play any special role in our arguments. Moreover in the next section we
will only consider groups expanded by automorphisms. On the one hand
we have included relations into our language in order to keep statements
as general as possible. On the other hand we believe that some form
of the results below concerning absolutely ubiquitous modules hold for
abelian structures in general. This suggests that the results of the paper
can be extended to expansions of groups by relations defining subgroups
in appropriate Gl.
3. Absolutely ubiquitous modules
In this section let (G, α¯) be an expansion of a group G by a tuple of
automorphisms. Consider the following situation. Assume that G has
a characteristic abelian subgroup H of finite index, |G : H| = m, and
H is a direct product of elementary abelian groups of infinite rank. Let
g1, ..., gm represent all cosets of H in G and let fi : H → H, i = 1, 2, ...,m,
be the group automorphism defined by fi(h) = g−1i hgi.
It is clear that for any β1, ..., βk ∈ Aut(H), the structure (H,β1, ..., βk)
can be considered as a Z[β1, ..., βk]/I-module on H, where I is the annu-
lator of Z[β1, ..., βk] on H. Moreover in the ω-categorical case H becomes
a module over a finite ring. This explains why we start this section with
a description of absolutely ubiquitous modules.
Any ω-categorical module over a finite ring R can be presented as
follows
M =
s
⊕
i=1
Bωi ⊕
t
⊕
i=1
Ckii ,
where s, t, ki ∈ ω and all Bi, Ci are indecomposable finite submodules.
This presentation of M is unique (see [1], [10]). The case M = Bω
64 Absolutely ubiquitous groups
with irreducible B (does not have proper non-trivial submodules) will be
principal in further arguments. Taking a quotient of R if necessary we
may assume that B is an exact R-module. It is well-known that in this
case R is simple and is isomorphic to the ring of all linear transformations
of the vector space B over the centraliser of B (see Sections 1-3 in [6]).
Proposition 3.1. Let R be any finite ring andM be an R-module
s
⊕
1
Bωi ⊕
t
⊕
1
Ckii , where Bi,Ci are indecomposable submodules. Then M is abso-
lutely ubiquitous if and only if all modules Bi are irreducible.
Proof. (⇒) Suppose that there is a number i such that Bi has a non-
trivial proper submodule A. Let components Bj with q < j ≤ s, be all
Bk properly imbeddable into Bi. In the case q 6= s consider the R-module
M ′ =
q
⊕
i=1
Bωi ⊕
t
⊕
i=1
Ckii .
In the case q = s let M ′ be the R-module
M ′ =
s
⊕
i=1
Bωi ⊕
t
⊕
i=1
Ckii ⊕A.
It is easy to see that J (M) = J (M ′) and M 6∼= M ′ (by ω-categoricity).
This contradicts absolute ubiquity.
(⇐) Assume that all Bi in the presentation ofM above are irreducible.
Let pi be the natural projection from M to its completely reducible part
s
⊕
i=1
Bωi .
Claim. For any extension E ⊆ F with F ∈ J (M) and
t
⊕
i=1
Ckii ≤ E there
is D such that F = D ⊕ E.
Consider the extension pi(E) ≤ pi(F ). By Section 4.1 of [6] any sub-
module of a completely irreducible module A is a direct summand in A.
Thus there is a submodule D < pi(F ) such that pi(F ) = D ⊕ pi(E). It is
easy to see that F = D ⊕ E.
The claim easily implies that any element of J (M) containing
t
⊕
i=1
Ckii
is of the form
s
⊕
i=1
Blii ⊕
t
⊕
i=1
Ckii .
Now assume that N is a module such that J (M) = J (N). We
have to construct an isomorphism F : M → N . Find an embedding
A. Ivanov, K. Majcher 65
τ0 :
t
⊕
i=1
Ckii → N and let
t
⊕
i=1
(C ′)kii be the image of τ0. By a back-
and-forth argument we extend τ0 to a sequence of partial isomorphisms
τi : Mi → Ni, i ∈ ω, from substructures of M to N .
Let M = {ai : i < ω} and N = {di : i < ω}. At Step 2l find the first
aj ∈M \M2l−1. By the claim above any proper extension of N2l−1 in N
is of the form D⊕N2l−1 where N2l−1 is of the form D′⊕
t
⊕
i=1
(C ′)kii . Then
D ⊕D′ is a completely reducible module and is of the form
s
⊕
i=1
Blii . For
an appropriate D (for example of the isomorphism type of
s
⊕
i=1
Blii with
large li) there is an embedding
τ2l : 〈M2l−1, aj〉 → D ⊕N2l−1
which extends τ2l−1.
The same argument works for Step 2l + 1 (by replacing the pair
M2l−1, N2l−1 by the pair N2l,M2l).
Lemma 3.2. Let M be an absolutely ubiquitous module
s
⊕
1
Bωi ⊕
t
⊕
1
Ckii ,
where Bi,Ci are indecomposable submodules and Bi are irreducible. Then
the expansion Th(M, {c : c ∈
t
⊕
1
Ckii }) admits elimination of quantifiers.
Proof. It suffices to show that every isomorphism over
t
⊕
1
Ckii between
finite substructures of M extends to an automorphism of M . Such an
automorphism is constructed as in the proof of Proposition 3.1.
We now return to the situation of groups H ≤ G described in the
beginning of the section. The following theorem develops Proposition 3.1
Theorem 3.3. If (G, α¯) is absolutely ubiquitous then (H, α¯, f¯) is abso-
lutely ubiquitous too.
Proof. For i, j ∈ {1, ..,m} let hij ∈ H be defined by the condition gi ·
gj = gk ·hij for k ∈ {1, ...,m}. In this case let gij denote gk. Suppose that
(H, α¯, f¯) is not absolutely ubiquitous. Then by ω-categoricity and Lemma
3.1 H has a decomposition into indecomposable submodules
s
⊕
i=1
Bωi ⊕
t
⊕
i=1
Ckii over Z[α¯, f1, ..., fm] where some Bi (which will be denoted by B)
has a nontrivial submodule A.
66 Absolutely ubiquitous groups
Fix a natural number l > ∑ti=1 ki. We can extend the multiplication
· to the product {g1, .., gm} ×H ×Al as follows
(gi, h, a1, .., al)·(gj , h′, a′1, .., a′l) = (gij , hij ·fj(h)·h′, fj(a1)·a′1, .., fj(al)·a′l).
As a result we obtain a group extension 0 → (H ⊕Al) → Gˆ→ (G/H) →
0. Any αi extends to an automorphism βi of Gˆ as follows:
βi((gj , h, a1, .., al)) = (αi(gj), αi(h), αi(a1), .., αi(al))
(since each fj is a conjugation it is easy to see that such βi is an auto-
morphism). It is clear that ({gi}i≤m ×H, β¯) ∼= (G, α¯).
In the case when A is isomorphic to some Bj , let H1 be a Z[α¯, f¯ ]-
submodule of H obtained from H by omitting all Bωj with Bj properly
embeddable into B, and then adding all hij . If A is not isomorphic to
any Bj then let H1 = H ⊕ Al. Let Gˆ1 be the subgroup of Gˆ generated
by H1 and {g1, ..., gm}.
Claim. J (Gˆ1, β¯) = J (G, α¯).
Since (G, α¯) is embeddable into (Gˆ, β¯) we have J (G, α¯) ⊆ J (Gˆ, β¯).
By the choice of H1 we have J (Gˆ, α¯) ⊆ J (Gˆ1, β¯).
It is clear that any structure from J (Gˆ1, β¯) can be embedded into a
structure of the form ({gi}i≤m ×H0 × Al, β) ∈ J (Gˆ1, β¯), where H0 is a
finite α¯f¯ -closed subgroup of H. Since (H0 × Al, β¯, f¯) naturally embeds
into (H, β¯, f¯) we see ({gi}i≤m ×H0 ×Al, β¯) ∈ J (G, α¯).
To see (Gˆ1, β¯) 6∼= (G, α¯) note that the intersection of H1 with any
α¯f¯ -closed subgroup of index m in Gˆ1 is an α¯f¯ -closed subgroup of index
≤ m in H1. In the case when H1 = H ⊕Al for sufficiently large l, such a
subgroup has suficiently many copies of A. Thus it cannot be a subgroup
of index ≤ m of a group isomorphic to H. In the remaining case this
statement is obvious. As a result we have that (G, α¯) is not absolutely
ubiquitous.
4. Absolutely ubiquitous groups
As in the previous sections let (G, α¯) be an expansion of a group G by
automorphisms. Assume that G has a characteristic abelian subgroup H
of finite index, |G : H| = m. Let g1, ..., gm represent all cosets of H in G
and let fi : H → H, i = 1, 2, ..,m, be the group automorphism defined
by fi(h) = g−1i hgi.
From now on we concentrate on the question which consequences
of absolute ubiquity of (G, α¯) become sufficient conditions for absolute
ubiquity of (G, α¯) under the assumption that (H, α¯, f¯) is absolutely ubiq-
uitous. We know that if (G, α¯) is absolutely ubiquitous then it is model
A. Ivanov, K. Majcher 67
complete. ThusH is defined by an existential formula φH(x) = ∃y¯φ1(x, y¯)
and the latter is equivalent in (G, α¯) to some universal formula ∀z¯φ2(x, z¯).
Let mH := max{|y¯|, |z¯|}.
On the other hand if (G, α¯) is absolutely ubiquitous then for all nat-
ural numbers r and s, any finite substructure C of (G, α¯) extends to a
finite substructure (D, α¯) ≤ (G, α¯) with the property that for any A ⊆ D
with |A| ≤ r and any set of formulas t(x¯) depending on an s-tuple of vari-
ables x¯ and with parameters from A, if t(x¯) has a realisation in (G, α¯)
then t(x¯) has a realisation in D (in this case we say that (D, α¯) is (r, s)-
saturated). This follows from so called Palyutin’s lemma presented in [5].
In fact the statement in [5] is much stronger: for an absolutely ubiquitous
M there is a function ρM (n) such that if a finite substructure D of M
contains copies of all ρ(r + s)-generated substructures of M then D is
(r, s)-saturated. We can now formulate the main result of the paper.
Theorem 4.1. Let (G, α¯) be a model complete and ω-categorical structure
such that
(i) there is a characteristic abelian subgroup H ≤ G of index m sat-
isfying the assumptions of the beginning of the section such that the cor-
responding expansion (H, α¯, f¯) is absolutely ubiquitous;
(ii) any finite substructure of (G, α¯) extends to a finite (1,mH)-satura-
ted substructure of (G, α¯).
Then (G, α¯) is absolutely ubiquitous.
We need some lemmas.
Lemma 4.2. Consider a structure (G′, α¯′) such that J (G′, α¯′) = J (G, α¯).
Let H ′ = ∀z¯φ2(G′, z¯), where φ2 is as above. Let D be an (1,mH)-
saturated substructure of G and D′ ⊆ G′ be a substructure of (G′, α¯′)
which is isomorphic to D under an isomorphism f : D → D′. Then
f(D ∩H) = D′ ∩H ′.
Proof. We start with the following claims.
Claim 1. (D, α¯) |= (∃y¯φ1(x, y¯) ↔ ∀z¯φ2(x, z¯)).
The implication ∃y¯φ1(x, y¯) → ∀z¯φ2(x, z¯) follows from the fact that
(G, α¯) |= ∃y¯φ1(x, y¯) → ∀z¯φ2(x, z¯). For the contrary implication apply
the fact that the structure (D, α¯) is (1,mH)-saturated. Thus for any
a ∈ D we have that if (G, α¯) |= ∃y¯φ1(a, y¯), then (D, α¯) |= ∃y¯φ1(a, y¯),
and if (D, α¯) |= ∀z¯φ2(a, z¯), then (G, α¯) |= ∀z¯φ2(a, z¯). This together with
assumptions on ∃y¯φ1(x, y¯) and ∀z¯φ2(x, z¯) implies the claim.
Claim 2. For any a′ ∈ D′, (D′, α¯′) |= ∀z¯φ2(a′, z¯) if and only if (G′, α¯′) |=
∀z¯φ2(a′, z¯).
It is obvious that (G′, α¯′) |= ∀z¯φ2(a′, z¯) implies (D′, α¯′) |= ∀z¯φ2(a′, z¯).
To see the converse suppose that there is a′ ∈ D′ such that (G′, α¯′) |=
68 Absolutely ubiquitous groups
¬(∀z¯φ2(a′, z¯)) and (D′, α¯′) |= ∀z¯φ2(a′, z¯). By Claim 1 the latter implies
(G′, α¯′) |= ∃y¯φ1(a′, y¯). As a result there are b¯′ and c¯′ such that (G′, α¯′) |=
¬φ2(a′, b¯′)∧φ1(a′, c¯′). Since the formula ¬φ2(x, z¯)∧φ1(x, y¯) does not have
solutions in G, we have a contradition with J (G′, α¯′) = J (G, α¯).
To finish the proof of the lemma we show that a ∈ H if and only
if f(a) ∈ H ′, where a ∈ D. If a ∈ H then (G, α¯) |= ∀z¯φ2(a, z¯),
(D, α¯) |= ∀z¯φ2(a, z¯) and (D′, α¯′) |= ∀z¯φ2(f(a), z¯). By Claim 2 (G′, α¯′) |=
∀z¯φ2(f(a), z¯) and so f(a) ∈ H ′.
If a 6∈ H then (G, α¯) |= ∀y¯¬φ1(a, y¯), (D, α¯) |= ∀y¯¬φ1(a, y¯) and by
Claim 1, (D, α¯) |= ¬∀z¯φ2(f(a), z¯). Thus (D′, α¯′) |= ¬∀z¯φ2(f(a), z¯) and
we have (G′, α¯′) |= ¬∀z¯φ2(f(a), z¯).
Now note that if a finite (1,mH)-saturated substructure D of (G, α¯)
contains more than m elements, then for any d1, ..., dm+1 ∈ D there
are i, j ≤ m + 1, i 6= j, such that (D, α¯) |= φH(did−1j ). Under the
circumstances of Lemma 4.2 we easily have that |G : H| = m = |G′ : H ′|.
Fix g′1, g′2, ..., g′m ∈ G′ representing all cosets of G′/H ′. Define f ′i(h′) :=
(g′i)−1h′g′i for all h′ ∈ H ′.
Lemma 4.3. Under the circumstances of Lemma 4.2 there exists a per-
mutation δ of the set {1, ...,m} such that
J (H, α¯, f1, ..., fm) = J (H ′, α¯′, f ′δ(1), ..., f ′δ(m)).
Proof. We present H and H ′ as ⋃{Ai : i ∈ ω} and
⋃{A′i : i ∈ ω}
where all Ai (resp. A′i) define an increasing sequence of finite α¯f¯ -closed
substructures of H (resp. finite α¯′f¯ ′-closed substructures of H ′). We
build two chains of embeddings of finite (1,mH)-saturated substructures
Di of G and D′i ≤ G′ , i ∈ ω, respectively (more precisely: substructures
D′i are isomorphic to (1,mH)-saturated substructures of G) such that
〈Ai, g1, ..., gn〉α¯ is embeddable into Di and 〈A′i, g′1, ..., g′n〉α¯
′ is embeddable
into D′i. Since J (G, α¯) = J (G′, α¯′) we can additionally arrange the
existence of a sequence of embeddings
D0 e0→ D′0
e′0→ D1 e1→ D′1
e′1→ ....
By Lemma 4.2 the embeddings ei, e′i induce a sequence of embeddings of
Z[α¯]- (resp. Z[α¯′]-)modules
φH(D0) → φH(D′0) → φH(D1) → φH(D′1) → ...→ ...
such that for every i the eie′i−1...e0(g¯)-conjugating on φH(D′i) agrees with
the e′ieie′i−1...e0(g¯)-conjugating under the embedding e′i and the corre-
sponding property holds for ei. Let σ′i be a permutation of set {1, ...,m}
A. Ivanov, K. Majcher 69
such that g′σ′i(j) · (eie
′
i−1...e0(gj))−1 ∈ φH(D′i), where g′i are taken by
the corresponding embedding of 〈A′i, g′1, ..., g′n〉α¯
′ into D′i. By the same
method define σi for φH(Di). Then all ei and e′i form a sequence of
embeddings of Z[α¯σi(f¯)]-(and Z[α¯′σ′i(f¯ ′)]-)modules. Choosing σ and σ′
which occur infinitely often as σi and σ′i we obtain a sequence of Z[α¯σ(f¯)]-
(and Z[α¯′σ′(f¯ ′)]-)modules. Now let δ = σ′σ−1. By the choice of Ai and
A′i we have
J (H, α¯, fσ(1), ..., fσ(m)) = J (limi φH(Di), α¯, fσ(1), ..., fσ(m)) =
J (lim
i
φH(D′i), α¯′, f ′σ′(1), ..., f ′σ′(m)) = J (H ′, α¯′, f ′σ′(1), ..., f ′σ′(m)).
As a result we have the conclusion of the lemma.
We can now finish the proof of the theorem. By Lemma 4.3 and
the first assumption of the theorem we may assume that the struc-
tures (H, α¯, f¯) and (H ′, α¯′, f¯ ′) are isomorphic and can be presented as
in Lemma 3.2. Let H =
s
⊕
1
Bωi ⊕
t
⊕
1
Ckii , where Bi,Ci are indecomposable
Z[α¯, f¯ ]-submodules and Bi are irreducible.
Let T = 〈g1, g2, ..., gm,
t
⊕
1
Ckii 〉α¯. Let T ⊆ Tˆ be a finite (1,mH)-
saturated substructure of (G, α¯). Find Tˆ ′ ⊆ G′ and an isomorphism I :
Tˆ → Tˆ ′. Let T ′ := I(T ). By Lemma 4.2 we see that I(T ∩H) = T ′ ∩H ′.
By Lemma 3.2 (that the expansion Th(M, {c : c ∈
t
⊕
1
Ckii }) ad-
mits elimination of quantifiers), the map I : T ∩ H → T ′ ∩ H ′ can
be extended to an isomorphism of Z[α¯, f¯ ]−(resp. Z[α¯′, f¯ I ]-)modules
I0 : H → H ′. Now the rule gih → I(gi)I0(h) with 1 ≤ i ≤ m and
h ∈ H, defines an isomorphism (G, α¯) → (G′, α¯′). For example, the
equality gih·gjh′ = gigjfj(h)h′ = gijhijfj(h)h′ corresponds to I(gi)I0(h)·
I(gj)I0(h′) = I(gi)I(gj)f ′j(I0(h))I0(h′) = I(gij)I(hij)f ′j(I0(h))I0(h′) =
I(gij)I0(hij)f ′j(I0(h))I0(h′).
We now formulate a theorem which summarises the material of the
paper. It follows from Theorem 2.1, Theorem 3.3 and Theorem 4.1.
Theorem 4.4. A model complete, ω-categorical group G is absolutely
ubiquitous if and only if G has a characteristic abelian subgroup H of
finite index such that the following conditions are satisfied.
(i) Let g1, ..., gm represent all cosets of H in G and let fi : H → H,
i = 1, 2, ...,m, be the group automorphism defined by fi(h) = g−1i hgi.
Then the Z[f¯ ]-module H is absolutely ubiquitous.
(ii) Let H be defined by an existential formula φH(x) = ∃y¯φ1(x, y¯)
and the latter is equivalent in G to some universal formula ∀z¯φ2(x, z¯).
70 Absolutely ubiquitous groups
Let mH := max{|y¯|, |z¯|}. Then any finite substructure of G extends to a
finite (1,mH)-saturated substructure of G.
We consider this theorem as some kind of algebraic description of ab-
solutely ubiquitous groups in the class of model complete ω-categorical
groups. Indeed condition (i) is algebraic by our description of absolutely
ubiquitous modules in Proposition 3.1. Condition (ii) looks more compli-
cated. Nevertheless it is a substantial reduction of the property arising
in Palyutin’s lemma from [5] (where there is no bound on m in the ap-
propriate condition of m-saturation).
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Contact information
A. Ivanov Institute of Mathematics, University of
Wroc law, pl.Grunwaldzki 2/4, 50-384
Wroc law, Poland
E-Mail: ivanov@math.uni.wroc.pl
K. Majcher Institute of Mathematics, University of
Wroc law, pl.Grunwaldzki 2/4, 50-384
Wroc law, Poland
E-Mail: majcher@math.uni.wroc.pl
Received by the editors: 01.02.2006
and in final form 21.11.2006.