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УДК 512.541
On Determinability of a Quotient Divisible Abelian Group of Rank 1 by Its Automorphism Group
Vadim K. Vildanov*
Lobachevsky State University of Nizhni Novgorod Gagarina, 23, Nizhni Novgorod, 603950
Russia
Egor A. Timoshenko^
Tomsk State University Lenina, 36, Tomsk, 634050 Russia
Received 13.02.2019, received in revised form 22.07.2019, accepted 20.09.2019 A criterion for determinability of a quotient divisible Abelian group of rank 1 by its automorphism group in the class of quotient divisible Abelian groups of rank 1 is considered in the paper.
Keywords: determinability, quotient divisible group, automorphism group. DOI: 10.17516/1997-1397-2019-12-6-699-704.
The word "group" will mean an Abelian group. As usual, A is a torsion group (a torsion-free group) if t(A) = A (t(A) = 0), where t(A) is the subgroup of all elements of finite order in A. A nonzero group A is a torsion-free group of rank 1 if it is isomorphic to a subgroup of q; we say that this group is a rank 1 torsion-free group of idempotent type if A is isomorphic to the additive group of some subring of q.
Let A £ X, where X is some class of groups. We say that A is determined by its automorphism group in the class X if Aut A = Aut B implies A = B for every B £ X.
It was proved that a p-group is determined by its automorphism group in the class of all p-groups for p > 5 [1]. Similar result was obtained for p = 3 [2]. Rank 1 torsion-free groups are not determined by their automorphism groups, but for direct sums of torsion-free groups of rank 1 (i.e., for completely decomposable torsion-free groups) necessary and sufficient conditions for determinability of a group by its automorphism group in some classes were found (see [3,4]). Among other things, the question of determinability of a group by its automorphism group was considered in the class of completely decomposable torsion-free groups whose direct summands of rank 1 have idempotent types [3]. In particular it was shown that a torsion-free group A of rank 1 and of idempotent type is determined by its automorphism group in the class of all such groups if and only if A is an infinite cyclic group. Let us note that the class of rank 1 torsion-free groups of idempotent type coincides with the class of quotient divisible groups of rank 1 which are torsion-free.
Let us denote the set of all primes by p. Recall (for example, see [5]) that a group A is p-divisible if the set pA = {pa | a £ A} coincides with A; a divisible group is a group which is p-divisible for all p £ p.
* kadirovi4@gmail.com t tea471@ mail.tsu.ru © Siberian Federal University. All rights reserved
Definition. Suppose that a group A does not contain nonzero divisible torsion subgroups and there is an element x € A of infinite order such that A/ (x) is a divisible torsion group. Then A is called the quotient divisible group of rank 1 and x is called the basis element of this group.
For an arbitrary quotient divisible group A of rank 1 we define a characteristic x = (mp)pEP by taking mp to be the smallest number m ^ 0 for which pmx is in pnA for all n > 0 (if there are no such numbers, we set mp = to). We say that x is the cocharacteristic of the element x. The cocharacteristic of a basis element of the group A is uniquely determined by A [6] and it is called the cocharacteristic of the group A.
The class of all quotient divisible groups of rank 1 is denoted by QD1. Let us recall the structure of quotient divisible groups of rank 1 (see [6,7]). Let x = (mp)pEP be an arbitrary characteristic and L be the set of all p € p for which 0 < mp < to. We denote the direct product of all rings Z/pmp z with p € L by Kx and we also denote the subring of the field q generated
by the elements - such that mp < to by qx (if all mp are infinite then qx = z). If L is a finite
xp x x
set, we define Rx = qx x Kx.
If the set L is infinite then Rx is the ring of all elements b = (bp)pEL € Kx such that for
u
some fraction — € qx the equality uep = vbp holds for almost all p € L, where ep is the identity
v
u
of the ring Z/pmp z). The mapping n that assigns to every such element b the number — is an
v
epimorphism of Rx onto qx. Its kernel Ker n coincides with the direct sum t(Rx) of all Z/pmpz with p € L.
It can be checked directly that for any characteristic x the additive group of the ring Rx is a quotient divisible group of rank 1, while the identity of the ring is the basis element of this group and it has x as its cocharacteristic. It follows that any quotient divisible group of rank 1 with cocharacteristic x is isomorphic to the additive group of the ring Rx [6]. In particular, if A € QV\ then the factor group of A with respect to its torsion part t(A) is a rank 1 group of idempotent type which is p-divisible for every p € p such that the p-component Ap is nonzero.
The symbol H denotes a restricted direct product of groups. This analogue of the notion of a direct sum is used for multiplicatively written groups.
Proposition 1. If A is a quotient divisible group of rank 1, then
Aut A = Aut(A/t(A)) x JJ Aut Ap, (*)
peL
where L is the set of all p € p for which Ap = 0.
Proof. We may assume that A coincides with one of the rings Rx. If L is finite then A is the direct sum of the p-components Ap with p € L and of the group qx = A/t(A) which is p-divisible for every p € L. Since each of these direct summands is fully invariant in A, we arrive at the isomorphism (*).
Now let the set L be infinite. It is known that any endomorphism of the additive group of the ring A = Rx is a multiplication by some element of A [6, 7]. From this we obtain Aut A = A*, where A* is the group of invertible elements of the ring A. It is easy to verify that the restriction p of the epimorphism n: A ^ qx to the group A* has the group (qx)* as its image. Its kernel Ker p is the set of elements b = (bp)pEL € A* such that bp = ep for almost all p € L.
Let us show that Ker p is a pure subgroup of A*. Suppose that b = (bp)pEL is an element of the group A* such that bn € Kerp (for some natural n). Let us define the element c = (cp)pEL € A*
by setting cp = bp if (bp)n = ep and cp = ep otherwise. Obviously cn = bn and c G Ker p as desired.
Imp = (qx)* is a restricted direct product of cyclic groups. Using Theorem 28.2 [5], we have A* = (qx)* x Ker p. Then
Aut A = (qx)* x Ker p = Aut qx x (Ap)* = Aut(A/t(A)) x Aut Ap.
peL peL
This completes the proof of the proposition. □
Cyclic groups of infinite order and of order n are denoted by Z and Z(n), respectively. It makes no difference whether these groups are written additively or multiplicatively.
Proposition 2. Suppose all p-components of a quotient divisible group A of rank 1 are nonzero. Then
Aut a = H z xH n I[z (qk )•
H0 qeP k>0 H0
Proof. It follows from the description of quotient divisible groups of rank 1 that A/t(A) = q. Using Proposition 1 and decomposing each group Aut Ap into the product of indecomposable cyclic subgroups, we obtain that
Aut a = n z x n n n z(qk),
Ho qeP k>0
where Jq,k < H0 for every q G p and k > 0.
We fix qk and consider the arithmetic progression with initial term 1 + qk and difference qk+1. By the Dirichlet theorem (see [8]) this progression contains infinitely many primes. If p is one of such primes and Ap = Z(pm) then Aut Ap = Z(p — 1) x Z(pm-1) has a direct factor Z(qk) since p — 1 is divisible by qk but not by qk+1. It follows that each of the cardinals Jq,k is equal to H0. This proves the proposition. □
Definition. We shall say that a finite cyclic group A is weakly determined by its automorphism group if for any finite cyclic group B ^ A the condition Aut A = Aut B implies that the number of nonzero Bp is greater than the number of nonzero Ap.
Let us note that if for a group A G QD1 and p G p we have Ap = 0 then group t(A) is p-divisible, and so the groups A and A/t(A) are simultaneously p-divisible or not p-divisible.
Theorem 3. A group A G QD1 is determined by its automorphism group in the class QD1 if and only if t(A) is a cyclic group (possibly equal to zero) which is weakly determined by its automorphism group and pA = A for all p G p such that Ap = 0.
Proof. A quotient divisible group of rank 1 with non-zero p-components is not determined by its automorphism group in the class QD1 because automorphism groups of all such quotient divisible groups are isomorphic to each other as it follows from Proposition 2.
Let us suppose that group A G QD1 has infinitely many nonzero p-components and its cocharacteristic x = (mp)per contains at least one symbol œ. Let us define ^ = (hp)peP = x by setting{
hp
if mp = œ,
p
m
p
Then for any group B € QD1 with cocharacteristic ^ we have B/t(B) = q and Bp = Ap for every p € p. Taking into account Proposition 1, it follows from the isomorphisms
Aut(A/t(A)) = Z(2) x Z = Autq = Aut(B/t(B))
«0
that Aut A = Aut B. Then group A is not determined by its automorphism group in QD1.
Let us assume that group B € QD1 has infinitely many nonzero p-components and its cocharacteristic ^ does not contain symbols to. Taking into account the case considered in the beginning of the proof, we may assume that ^ contains at least one symbol 0. Then ^ can be obtained from some characteristic x = ¥ using the procedure described in the previous paragraph. Therefore for every group A € QD1 with cocharacteristic x we have A £ B and Aut A = Aut B. Hence B is not determined by its automorphism group in the class QD1.
It remains to consider the case when the group A € QD1 has a finite number of nonzero p-components, i.e., when the group t(A) is cyclic (then A splits). It follows from Proposition 1 and from the uniqueness (up to isomorphism) of the decomposition of a group into the restricted direct product of indecomposable cyclic groups that isomorphism Aut A = Aut B with B € QD1 is possible only in the case when t(B) is a cyclic group, Aut(A/t(A)) = Aut(B/t(B)) and Aut(t(A)) = Aut(t(B)). We denote the set of all p € p for which Ap = 0 by L and the set of all p € p for which the group A/t(A) is p-divisible by X (then L c X).
1) Suppose that equality Ap =0 does not imply pA = A, i.e., that set X strictly contains L. Let us choose a set Y c p so that L c Y, Y = X and \Y\ = \X|. If R is the subring of the field q generated by elements 1 such that q € Y then for the group B = R © t(A) € QD1 we have A £ B (because A/t(A) £ R) and
Aut(B/t(B)) = Aut R = Z(2) x JJ Z = Aut(A/t(A)),
I* I
whence we obtain Aut B = Aut A. This means that the group A is not determined by its automorphism group in the class QD1.
2) Now assume that X = L but the group t(A) is not weakly determined by its automorphism group. Then there is a finite cyclic group G £ t(A) such that Aut G = Aut(t(A)), and the number of non-zero primary components of G does not exceed \L\. Let us choose a set Y c p so that it contains all q € p for which Gq = 0 and so that \Y\ = \L\. If R is the subring of the field q generated by elements 1 with q € Y then for the group B = R © G € QD1 we have B £ A and Aut(B/t(B)) = Z(2) x ZILI = Aut(A/t(A)), whence we obtain Aut B = Aut A. Consequently, in this case the group A is again not determined by its automorphism group in QD1.
3) Finally, suppose that X = L and the group t(A) is weakly determined by its automorphism group. Let Aut A = Aut B with B € QD1. Then it follows from the isomorphism Aut(A/t(A)) = Aut(B/t(B)) that the set of all q € p such that B/t(B) is a q-divisible group has cardinality \L\. This means that B has at most \L\ non-zero primary components. Then Aut(t(A)) = Aut(t(B)) implies the isomorphism t(A) = t(B). Thus for every p € L we have Bp = 0, and hence B/t(B) is a p-divisible group. Therefore A/t(A) = B/t(B) and so A = B. We obtain that group A is determined by its automorphism group in the class QD1. The theorem is proved. □
The theorem just proved implies that a quotient divisible group of rank 1 with cocharacteristic
x = (m2,mз,m5,... )
is determined by its automorphism group in the class QD1 if and only if x does not contain symbols 0 and mp = to for almost all p £ p. The finite mp defines a cyclic group which is weakly determined by its automorphism group.
It can be verified directly that if n ^ 50 then Z(n) is weakly determined by its automorphism group for n = 1, 5, 8, 11, 13, 16, 17, 23, 24, 25, 29, 31, 32, 37, 41, 47. From this we immediately obtain examples of quotient divisible groups of rank 1 which are determined by their automorphism groups in the class QD1. For instance, at n = 5 and n = 24 we have quotient divisible groups of rank 1 with cocharacteristics (to, to, 1, to, ..., to, ...) and (3,1, to, to, ..., to, ...), respectively.
The work of the second author was supported by the Ministry of Science and Higher Education of Russia (assignment no. 1.13557.2019/13.1).
References
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[3] V.K.Vildanov, Determinability of a completely decomposable torsion-free Abelian group of rank 2 by its automorphism group, Vestn. Nizhegorod. Univ. N. I. Lobachevskogo, (2011), no. 3(1), 174-177 (in Russian).
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X. Vildanov, Egor Л. Timoshenko Си Determinability of a Quotient В1У181Ъ1С Abelian Group .
Об определяемости факторно делимой абелевой группы ранга 1 своей группой автоморфизмов
Вадим К. Вильданов
Нижегородский государственный университет им. Лобачевского Гагарина, 23, Нижний Новгород, 603950
Россия
Егор А. Тимошенко
Томский государственный университет Ленина, 36, Томск, 634050 Россия
Получен критерий определяемости факторно делимой абелевой группы 'ранга 1 ее группой автоморфизмов в классе факторно делимых абелевых групп ранга 1.
Ключевые слова: определяемость, факторно делимая группа, группа автоморфизмов.
