Some new facts about group 𝒢 generated by the family of convergent permutations

Abstract The aim of this paper is to present some new and essential facts about group 𝒢 generated by the family of convergent permutations, i.e. the permutations on ℕ preserving the convergence of series of real terms. We prove that there exist permutations preserving the sum of series which do not belong to 𝒢. Additionally, we show that there exists a family G (possessing the cardinality equal to continuum) of groups of permutations on ℕ such that each one of these groups is different than 𝒢 and is composed only from the permutations preserving the sum of series. This result substantially strengthens some old Pleasants’ result.


Introduction
The family of all permutations on N will be denoted by P: A permutation p 2 P is called a convergent permutation if for every convergent series P a n of real terms the p rearranged series P a p.n/ is convergent as well. A family of all convergent permutations will be denoted by C: If we replace in the definition of convergent permutation p series P a n of real terms by series P v n of vector terms, and even by series of normed abelian semigroup terms, the respective set of all convergent permutations will be the same as in the real series case. As Stoller [1] observed for some normed abelian groups (e.g. the p-adics) the set of permutations preserving convergence is larger than C. Permutations belonging to the family D WD P n C will be called the divergent permutations. A group of permutations generated by C will be denoted by G: We know that G ¤ P (Pleasants [2,3], see also Corollary 4.2 in Section 4).
Let A; B 2 P: Then the following family of permutations of N fp 2 P W p 2 A and p 1 2 Bg will be denoted by AB: After Kronrod [4] and Wituła we call a) elements of CC -the two-sided convergent permutations, b) elements of CD -the one-sided convergent permutations, c) elements of DC -the one-sided divergent permutations, d) elements of DD -the two-sided divergent permutations.
The symbol ı denotes here the composition of subsets of P; i.e.
B ı A D fq ı p. / WD q.p. // W q 2 B and p 2 Ag for any nonempty subsets A; B of P. We say that a set A P is algebraically big if A ı A D P. For example, the family DD is only the double-sided family of permutations defined in a)-d), which is algebraically big. Moreover, we note that (see [5,6]): : : We think that all these inclusions are really strict. We are working now on some details of the proof of this conjecture. For the shortness of notation we will write qp (qp.n/, respectively) instead of q ı p (q ı p.n/, respectively). By S we denote the family of all permutations on N preserving the sum of series, it means permutations p 2 P such that if P a n is the convergent series with real terms and the p-rearranged series P a p.n/ is also convergent then P a n D P a p.n/ , i.e. the sums of both series are the same 1 . The set S is also algebraically big (Stoller [1]).
More precisely Stoller proved that the set D.1/ WD˚p 2 P W p.f1; 2; : : : ; ng/ D f1; 2; : : : ; ng for infinitely many n 2 N « is algebraically big and we have D.1/ S [6]. It will be proven in Section 4 that there exists the family G (possessing the cardinality equal to continuum) of groups of permutations on N such that for every G 2 G we have G S and G n G ¤ ;.
All subsets of family P, introduced here, are nonempty. Some basic algebraic type properties of all these and some others subsets of P are discussed in papers [5][6][7][8][9][10].

Technical notations and auxiliary results
Let A be a nonempty subset of N, let fa n g denote the increasing sequence of all elements of A. Subset I of A is said to be an interval of A, providing that it has the form fa n ; a nC1 ; :::; a nCm g for some m 2 N [ f0g and n 2 N. So, only the nonempty intervals are discussed here.
We say that two intervals I and J of the given nonempty subset A of N are separated if the set I [ J is not an interval of A.
Let A be a nonempty subset of N and let B be a nonempty subset of A. Then the notation set B is a union of n (or of at most n, or of at least n, respectively) MSI(A) means that there exists a family I of n (or of at most n, or of at least n, respectively) intervals of set A such that each two intervals are separated. In short we say that the family I form n (or of at most n, or of at least n, respectively) mutually separated intervals of A.
For the given nonempty set A Â N we will denote by S.A/ the set of all permutations of set A: Certainly we have P D S.N/: In particular, S n WD S.f1; 2; :::; ng/ denotes the symmetric group of degree n for every n 2 N. 1 Symbol P a n , depending on the context of considerations, denotes either the respective series, i.e. the sequence of partial sums ( , or the sum of this series if it is convergent Next, for every p 2 P and an infinite set A N by c.pj A / we denote supfn 2 N W there exists an interval B of A such that the set p.B/ is a union of n MSI(p.A/) g: (1) Instead of c.pj N / we will write c.p/ for every p 2 P. We know (see [11,12]) that permutation p 2 P is a convergent permutation if and only if c.p/ 2 N (for the other characterizations of convergent permutations on N see also [13]). Furthermore, if A N and both sets A and its complement are infinite then for every p 2 P such that p.A/ D A we get that p is a convergent permutation on A (which by definition means that if the given real series P n2A a n is convergent then also the respective p-rearranged series P n2A a p.n/ is convergent) if and only if c.pj A / 2 N (the value of c.pj A / is defined by (1)). The family of all convergent permutations p 2 P on A such that p.A/ D A will be denoted by C.A/. The complement of C.A/ to the set fp 2 P W p.A/ D Ag will be denoted by D.A/.
The symbol P .p/ denotes the convergence class of permutation p, i.e. the family of all convergent real series P a n for which the p-rearranged series P a p.n/ is also convergent. We note that the following fundamental results hold true. .p/ Â P .q/ or P .p/ n P .q/ ¤ ; and P .q/ n P .p/ ¤ ; (we say in this case that p and q are incomparable). Moreover, the respective permutations p; q could be always chosen to be incomparable.
At last, if there exist two disjoint infinite sets A; B N such that c.pj p 1 .A/ / < 1; c.qj q 1 .A/ / D 1; c.pj p 1 .B/ / D 1 and c.qj q 1 .B/ / < 1 then the permutations p and q are incomparable. If additionally we suppose that there exist a partition fI n g 1 nD1 of N, created from the successive intervals of N, such that p.I n / D q.I n / D I n for every n 2 N (which implies that p and q belong to D.1/) and if there exists an infinite set A Â N such that and supfk W there exists an interval I of S n2NnA I n such that each of the sets p.I / n q.I / and q.I / n p.I / is a union of at most k MSIg < 1; (2) then P .p/ P .q/ (the strict inclusion holds here).
Detailed proof of Theorem 2.1 will be omitted here since it can be easily deduced from the definitions of concepts used in here.
We would like to make one more remark. The Reader can feel a great deficiency about the sufficient conditions given in the last part of Theorem 2.1, which implies that inclusion P .p/ P .q/ can be satisfied (however these conditions are very useful in analysis of permutations presented in other places of the paper). We will additionally respond to these doubts in such a way that it is impossible to obtain this inclusion under very general assumptions, because in paper [15] we will present the example of set A N such that this set and its complement are both infinite and of the respective permutations p; q 2 D such that c.pj p 1 .A/ / < 1; c.qj q 1 .A/ / D 1; c.pj p 1 .NnA/ / < 1 and c.qj q 1 .NnA/ / < 1: Nevertheless, permutations p and q are incomparable. Definition 2.4. By the symbol S n .k; b/, for b; n; k 2 N; we will denote the family of all permutations p 2 S n which can be presented in the form of composition of k permutations belonging to S n , it means in the form p D q k q k 1 :::q 1 , where q i 2 S n ; i D 1; 2; :::; k; and, simultaneously, The following lemma, coming from paper [2], gives an estimation of the number of b-connected permutations belonging to group S n .
Lemma 2.5. The number of b-connected permutations belonging to group S n is majorized by .8b ln n/ 2bn . In the sequel we obtain card S n .k; b/ Ä 2 .8 b ln n/ 2bnk : 3 Some algebraic facts on the family P n G Let us start with two results of technical character.
Lemma 3.1. Let G P be a group of permutations. If q; p; p n 2 P n G for some n 2 N; n 2; then for every k 2 N; k < n; we have either p k q 2 P n G or p n k q 2 P n G: Proof. Suppose that for some k < n we have p k q 2 G and p n k q 2 G: Hence, also p n D .p n k q/.q 1 p k / D .p n k q/.p k q/ 1 2 G which is impossible.
Lemma 3.2. Let G P be a group of permutations. Then the following relations hold true i) .P n G/ 1 D P n G; ii) fgg ı .P n G/ D .P n G/ ı fgg D P n G for every g 2 G; iii) G .P n G/ ı .P n G/: iv) If there exists g 2 P n G such that also g 2 2 P n G, then P n G is the algebraically big subset of P: Moreover, for any '; 2 P n G at least one of the compositions g', g or ' also belongs to P n G. Furthermore, either g' 2 P n G or g 1 ' 2 P n G, and either 'g 2 P n G or 'g 1 2 P n G.
Proof. Since G is a group, therefore relations i / and i i / are obvious. Let 2 G, g 2 P n G: Then also g 1 2 P n G which implies that D . g/g 1 2 .P n G/ ı .P n G/; i.e. relation i i i / holds. If ' 2 P n G and additionally g 2 2 P n G; then by Lemma 3.1 either 'g 2 P n G or 'g 1 2 P n G which implies ' D .'g/g 1 D .'g 1 /g 2 .P n G/ ı .P n G/, i.e. P n G .P n G/ ı .P n G/ and by i i i / relation P D .P n G/ ı .P n G/ holds. Suppose that g', g and ' all belong to G. Then also which gives the contradiction. We prove that by item iv/ of Lemma 3.2 the set P n G is the algebraically big subset of P. To this aim we need a special case (k D 3) of the following intriguing result.
Theorem 3.4. For every k 2 N; k > 1; equation p k D id N possesses the solution p 2 P n G such that p i 2 P n G for every i D 1; 2; :::; k 1: Proof. From Lemma 2.5 and the Stirling formula [14] we get that there exists an increasing sequence of natural numbers fn.u/ W u 2 Ng such that S n.u/ n S n.u/ .u; u/ ¤ ; for every u 2 N: Let e p u 2 .S n.u/ n S n.u/ .u; u// for every u 2 N: Let us fix k 2 N; k > 1; and let fI u g be the increasing sequence of intervals (which means that i < j for any i 2 I u , j 2 I uC1 , u 2 N) forming a partition of N and satisfying condition card I u D k n.u/; for every u 2 N: Let us put J u D OE.k 2/ n.u/ C min I u ; .k 1/ n.u/ C min I u for u 2 N. The expected permutation p is defined in the following way p.i n.u/ C j C min I u / D .i C 1/ n.u/ C j C min I u ; p..k 2/n.u/ C j C min I u / D .k 1/ n.u/ C e p u .j C 1/ 1 C min I u ; p..k 1/n.u/ C j C min I u / D e p 1 u .j C 1/ 1 C min I u ; for i D 0; 1; :::; k 3 and j D 0; 1; :::; n.u/ 1: The verification of condition p k D id N is trivial, whereas the fact that p i 2 .P n G/ for every i D 1; 2; :::; k 1 is a consequence of the fact that composition of the following three mappings: the increasing mapping of interval OE1; n.u/ onto interval J u , the restriction of p to interval J u and the increasing mapping of interval OE.k 1/ n.u/ C min I u ; max I u onto interval OE1; n.u/, is equal to e p u for every u 2 N: Now, let us suppose that there exist permutations p i 2 P; i D 1; 2; :::; n; such that p D p n p n 1 :::p 1 and either p i 2 C or p 1 i 2 C; for every i D 1; 2; :::; n: Let J be an interval of natural numbers. We denote by p J and p i;J the restrictions of permutations p and p i , respectively, to sets J and p i 1 p i 2 :::p 1 p 0 .J /; respectively, for each i D 1; 2; :::; n; where p 0 WD id N . Then the following decomposition holds q n;J p j q 1 1;J D .q n;J p n;J q 1 n 1;J /.q n 1;J p n 1;J q 1 n 2;J /:::.q 2;J p 2;J q 1 1;J /.q 1;J p 1;J q 1 0;J /; where q i;J ; 1 Ä i Ä n; is the increasing mapping of set p i p i 1 :::p 1 .J / onto interval OE1; card.J / and q 0;J WD id N : Let us notice that for each i D 1; 2; :::; n we have q i;J p i;J q 1 i 1;J 2 S card.J / and this is the b-connected permutation for b D c.p i / or b D c.p 1 i /, depending on the fact whether p i 2 C or p 1 i 2 C: Justification is needed only for the last property.
So let us suppose that there exists 1 Ä i Ä n such that p i 2 C and simultaneously D q i;J p i;J q 1 i 1;J is not the b-connected permutation. Then there exists subinterval ƒ of interval OE1; card.J / such that the set .ƒ/ is a union of at least .b C 1/ MSI. Let I be the family of mutually separated intervals of natural numbers forming the decomposition of set .ƒ/: Auxiliarly we takę D min q 1 i 1;J .ƒ/ andˇD max q 1 i 1;J .ƒ/: Then from the fact that q 1 i 1;J is the increasing mapping we get the following relations Let us notice one more property. For any ‚; " 2 I; ‚ ¤ "; if q 1 i;J .‚/ < q 1 i;J ."/ then the following relation holds (6) .max q 1 i;J .‚/; min q 1 i;J ."// \ p i q 1 i;J .OE1; card.J // ¤ ;: From the relations (4), (5) and (6) we are able to deduce that the set p i .OE˛;ˇ/; as well as the set .ƒ/; is the union of at least .b C1/ MSI, since the numbers from the set p i .OE˛;ˇnq 1 i 1;J .ƒ// "do not fill" completely all the "holes" between sets q 1 i;J .‚/ and q 1 i;J ."/ for any ‚; " 2 I; ‚ ¤ ": Thereby we get the contradiction with definition of number b and, in consequence, with the assumption that is not the b-connected permutation.
Thus, if we assume that v > n C max i otherwiseg then p u D q n;J p J q 1 1;J … S n.u/ .u; u/ for J D J u and for every u 2 N; u > v; which contradicts the assumption.
Corollary 3.5. The set P n G is algebraically big.
Referring to Theorem 3.4, as well as to item iv/ of Lemma 3.2, we present one more important result.
Theorem 3.6. Let p 2 P n G. If also p 2 2 P n G, then there exists a family B p P n G such that card B p D c and for every q 2 B p we have pq 2 C and qp 2 C: The proof of this result is based on the following unexpected result, a proof of which will be presented in paper [16], since we have received this result exactly in the course of preparing this paper. Thus, the proof of Theorem 3.6 is the next point of its application. Proof of Theorem 3.6. By item i / of Lemma 3.2 we have p 1 2 P n G: Thus, from Theorem 3.7 there exists a family A p 1 C C possessing the respective properties i / i i i / given above. In the sequel, if . ; ı/ 2 A p 1 then p 1 D p 1 ı 2 P n G, because of item i i / of Lemma 3.2 and p.p 1 ı/ D ı 2 C and . p 1 /p D 2 C: It is sufficient to set B p WD˚ p 1 W . ; ı/ 2 A p 1 « : We get card B p D c because of the condition i i i / of Theorem 3.7.
Let us finish this section with one more result concerning the semigroups with one generator.
Proposition 3.8. Let G S P. Let us suppose that G is a group of permutations and S is a semigroup of permutations such that .S n G/ 1 \ S D ;: Then for any two p; q 2 S n G the relation pq 2 S n G is true as well.
In the sequel, we obtain fp n W n 2 Ng S n G and fp n W n 2 Ng P n S .
Proof. Let p; q 2 S n G. If pq 2 G then p 1 D q.pq/ 1 2 S which is impossible. Similarly, if p n 2 S for some n 2 N then also p 1 D p n 1 .p n / 2 S which again contradicts the assumptions. where symbol Fin denotes the family of all almost everywhere identical permutations p 2 P; whereas F (see [7]) denotes the family of all permutations p 2 P for which there exists a finite partition N 1 ; N 2 ; : : : ; N k of set N such that all restrictions pj N i , i D 1; 2; : : : ; k are the increasing mappings. We note that C \ .P n F/ ¤ ;, i.e., G \ .P n F/ ¤ ; (the respective example can be easily created).

Pleasants' result
Let fn k g 1 kD1 be an increasing sequence of positive integers such that lim sup k!1 .n kC1 n k / D 1 and n 1 D 1: Then we denote by G.fn k g 1 kD1 / the family of all permutations p 2 P defined in the following way pj OEn k ;n kC1 / 2 S n kC1 n k ; for every k 2 N, which by definition means that p.OEn k ; n kC1 // D OEn k ; n kC1 / and q k 2 S n kC1 n k ; where q k .i / WD p.n k C i 1/ for every i D 1; 2; :::; n kC1 n k and k 2 N: We note the following facts.
The following last result of this paper could be deduced from the proof of Theorem 4.1 on the basis of its proof.
Theorem 4.5. There exists a family fG x W x 2 Rg of subgroups of P satisfying conditions i / iv/ of Theorem 4.1. Moreover, for every pair x; y 2 R; x ¤ y; there exist the subsets G 0 x G x and G 0 y G y , both having the power of continuum and such that any two permutations p 2 G 0 x and q 2 G 0 y are incomparable which means that the following relations hold X .p/ n X .q/ ¤ ; and X .q/ n X .p/ ¤ ;: Proof. First we fix a Sierpiński's family S of increasing sequences of positive integers. We can also suppose that if fn k g 1 kD1 2 S then lim k!1 .n kC1 n k / D 1. Next, with each fn k g 1 kD1 2 S we connect a family G 0 fn k g 1 kD1 G fn k g 1 kD1 defined in the following way. Firstly, for every divergent permutation p on N we set q.n k / D n p.k/ ; k 2 N; and qj I D id I ; where I D N n fn k g 1 kD1 : Secondly, let us define G 0 to be the family of all permutations q defined in this way. Certainly we have qj fn k g 1 kD1 2 D fn k g 1 kD1 for every fn k g 1 kD1 2 S and permutation p 2 D, and since any two different sequences fn k g 1 kD1 and fm l g 1 lD1 from S are almost disjoint we get the relations X .'/ n X .q/ ¤ ; and X .q/ n X .'/ ¤ ; which hold for any two q 2 G 0 fn k g 1 kD1 and ' 2 G 0 fm l g 1 lD1 . Both these families G 0 possess the power of continuum since card D D c.
The above relations from the Theorem 2.1 can be deduced.

Reflection about Theorem 4.5
Why this result is so important to us? Because in our considerations we do not want to get away from the roots of our research. We succeeded in this theorem to give the analytical weight to the distinguished subgroups of group P differentiating these subgroups. It is certainly the incomparability relation of permutations related to their convergence classes. We think that this phenomenon is valuable because the excessive algebraization of this research may deprive it of the natural power and inquisitiveness resulting from its connection with the theory of series.