Denote the sum of element orders in a finite group G by ψ(G) and let C n denote the cyclic group ... more Denote the sum of element orders in a finite group G by ψ(G) and let C n denote the cyclic group of order n. Suppose that G is a non-cyclic finite group of order n and q is the least prime divisor of n. We proved that ψ(G) The first result is best possible, since for each n = 4k, k odd, there exists a group G of order n satisfying ψ(G) = 7 11 ψ(C n ) and the second result implies that if G is of odd order, then ψ(G) < 1 2 ψ(C n ). Our results improve the inequality ψ(G) < ψ(C n ) obtained by H. Amiri, S.M. Jafarian Amiri and I.M. Isaacs in 2009, as well as other results obtained by S.M. Jafarian Amiri and M. Amiri in 2014 and by R. Shen, G. Chen and C. Wu in 2015. Furthermore, we obtained some ψ(G)-based sufficient conditions for the solvability of G.
Denote by G a finite group and let ψ(G) denote the sum of element orders in G. In 2009, H.Amiri, ... more Denote by G a finite group and let ψ(G) denote the sum of element orders in G. In 2009, H.Amiri, S.M.Jafarian Amiri and I.M.Isaacs proved that if |G| = n and G is non-cyclic, then ψ(G) < ψ(C n ), where C n denotes the cyclic group of order n. In 2018 we proved that if G is non-cyclic group of order n, then ψ(G) ≤ 7 11 ψ(C n ) and equality holds if In this paper we proved that equality holds if and only if n and G are as indicated above. Moreover we proved the following generalization of this result: Theorem 4. Let q be a prime and let G be a non-cyclic group of order n, with q being the least prime divisor of n. Then ψ(G) ≤ ((q 2 -1)q+1)(q+1) q 5 +1 ψ(C n ), with equality if and only if n = q 2 k with (k, q) = 1 and G = (C q × C q ) × C k . Notice that if q = 2, then ((q 2 -1)q+1)(q+1) q 5 +1 = 7 11 .
Annali di Matematica Pura ed Applicata, Dec 1, 1986
Let 5~ be the class of groups de]i/ned by : G e J~ if and only if H ~ ~ is finitely generated whe... more Let 5~ be the class of groups de]i/ned by : G e J~ if and only if H ~ ~ is finitely generated whenever H and K are finitely generated subgroups of G. In tMs paper we get the following theorems: -Let G be a nilpotent by abelian group satisfying the maximal condition on normal subgroups. Then the ]ollowing statements are equivalent: (2) every finitely generated subgroup o/ G is polyeyelie or of finite index in G; (3) G ~ is polycyclie and G/G" e J~.
Denote by G a finite group and by ψ(G) the sum of element orders in G. If t is a positive integer... more Denote by G a finite group and by ψ(G) the sum of element orders in G. If t is a positive integer, denote by C t the cyclic group of order t and write ψ(t) = ψ(C t ). In this paper we proved the following Theorem A: Let G be a non-cyclic group of odd order n = qm, where q is the smallest prime divisor of n and (m, q) = 1. Then the following statements hold. (1) If q = 3, then 301 , and equality holds if and only if n = 3 , where p is the smallest prime bigger than q and equality holds if and only if n = qp 2 m 1 with (m 1 , p!) = 1 and G = C q × C p × C p × C m 1 .
Journal of Algebra and Its Applications, Feb 17, 2019
In this note we present a new subgroup embedding property that can be considered as an analogue o... more In this note we present a new subgroup embedding property that can be considered as an analogue of pronormality in the scope of permutability and Sylow permutability in finite groups. We prove that finite PST-groups, or groups in which Sylow permutability is a transitive relation, can be characterised in terms of this property, in a similar way as T-groups, or groups in which normality is transitive, can be characterised in terms of pronormality.
Journal of the Australian Mathematical Society, Aug 1, 1992
Groups in which every infinite set of subgroups contains a pair that permute were studied by M. C... more Groups in which every infinite set of subgroups contains a pair that permute were studied by M. Curzio, J. Lennox, A. Rhemtulla and J. Wiegold. The question whether periodic groups in this class were locally finite was left open. Here we show that if the generators of such a group G are periodic then G is locally finite. This enables us to get the following characterisation. A finitely generated group G is centre-by-finite if and only if every infinite set of subgroups of G contains a pair that permute.
Following J.S. Rose, a subgroup H of a group G is said contranormal in G if G = H G . In a certai... more Following J.S. Rose, a subgroup H of a group G is said contranormal in G if G = H G . In a certain sense, contranormal subgroups are antipodes to subnormal subgroups. It is well known that a finite group is nilpotent if and only if it has no proper contranormal subgroups. We prove that a nilpotent-by-finite group with no proper contranormal subgroup is nilpotent. There are locally nilpotent groups with a proper contranormal subgroup. We study the structure of hypercentral groups with a finite proper contranormal subgroup.
Proceedings of the American Mathematical Society, Oct 25, 1999
of subgroups such that for each ordinal α, Gα G, and G α+1 /Gα has no nonabelian free subsemigrou... more of subgroups such that for each ordinal α, Gα G, and G α+1 /Gα has no nonabelian free subsemigroup, then G is right orderable if and only if it is locally indicable. In particular if G is a radical-by-periodic group, then it is right orderable if and only if it is locally indicable.
In this paper we shall deal with periodic groups, in which each element has a prime power order. ... more In this paper we shall deal with periodic groups, in which each element has a prime power order. A group G will be called a BCP -group if each element of G has a prime power order and for each p ∈ π(G) there exists a positive integer u p such that each p-element of G is of order p i ≤ p u p . A group G will be called a BSP -group if each element of G has a prime power order and for each p ∈ π(G) there exists a positive integer v p such that each finite p-subgroup of G is of order p j ≤ p v p . Here π(G) denotes the set of all primes dividing the order of some element of G. Our main results are the following four theorems. Theorem 1: Let G be a finitely generated BCP -group. Then G has only a finite number of normal subgroups of finite index. Theorem 4: Let G be a locally graded BCP -group. Then G is a locally finite group. Theorem 7: Let G be a locally graded BSP -group. Then G is a finite group. Theorem 9: Let G be a BSP -group satisfying 2 ∈ π(G). Then G is a locally finite group.
We study an inverse problem of small doubling type. We investigate the structure of a finitely ge... more We study an inverse problem of small doubling type. We investigate the structure of a finitely generated group $G$ such that, for any set $S$ of generators of $G$ of minimal order, we have $S^2 leq 3|S|-beta$, where $beta in {1, 2, 3}$
In this note, we present a new subgroup embedding property that can be considered as an analogue ... more In this note, we present a new subgroup embedding property that can be considered as an analogue of pronormality in the scope of permutability and Sylow permutability in finite groups. We prove that finite PST-groups, or groups in which Sylow permutability is a transitive relation, can be characterized in terms of this property, in a similar way as T-groups, or groups in which normality is transitive, can be characterized in terms of pronormality.
Suppose that 12 is a variety of groups defined by means of a law v(x•,... ,x,•) = 1, and say that... more Suppose that 12 is a variety of groups defined by means of a law v(x•,... ,x,•) = 1, and say that a group G belongs to the class 12' if for any infinite subsets X•,... ,X,• there exist elements xi • Xi, for all i • {1,... ,n}, such that V(Xl,... ,x,•)= 1. Obviously the class 12 and the
Journal of the Australian Mathematical Society, 1992
Groups in which every infinite set of subgroups contains a pair that permute were studied by M. C... more Groups in which every infinite set of subgroups contains a pair that permute were studied by M. Curzio, J. Lennox, A. Rhemtulla and J. Wiegold. The question whether periodic groups in this class were locally finite was left open. Here we show that if the generators of such a group G are periodic then G is locally finite. This enables us to get the following characterisation. A finitely generated group G is centre-by-finite if and only if every infinite set of subgroups of G contains a pair that permute.
... Since the g i i are free generators for K modulo Z, we have KH KZ; also K/ KZ is free abelian... more ... Since the g i i are free generators for K modulo Z, we have KH KZ; also K/ KZ is free abelian. But ZCG HCG K , since centralizers are isolated, and so ZKZ K. The result follows. Patrizia Longobardi, Mercede Maj and Howard Smith 292 Page 3. ...
ABSTRACT Let D i be the class of groups with i conjugacy classes of infinite size and let D be th... more ABSTRACT Let D i be the class of groups with i conjugacy classes of infinite size and let D be the union of all D i for all i=0,1,2,⋯. In these terms D 0 is exactly the class of FC-groups. In this article the authors investigate the D-groups. They obtain some interesting properties of these groups, characterize periodic locally graded non-FC-groups, and study in detail D 1 -groups.
Denote the sum of element orders in a finite group G by ψ(G) and let C n denote the cyclic group ... more Denote the sum of element orders in a finite group G by ψ(G) and let C n denote the cyclic group of order n. Suppose that G is a non-cyclic finite group of order n and q is the least prime divisor of n. We proved that ψ(G) The first result is best possible, since for each n = 4k, k odd, there exists a group G of order n satisfying ψ(G) = 7 11 ψ(C n ) and the second result implies that if G is of odd order, then ψ(G) < 1 2 ψ(C n ). Our results improve the inequality ψ(G) < ψ(C n ) obtained by H. Amiri, S.M. Jafarian Amiri and I.M. Isaacs in 2009, as well as other results obtained by S.M. Jafarian Amiri and M. Amiri in 2014 and by R. Shen, G. Chen and C. Wu in 2015. Furthermore, we obtained some ψ(G)-based sufficient conditions for the solvability of G.
Denote by G a finite group and let ψ(G) denote the sum of element orders in G. In 2009, H.Amiri, ... more Denote by G a finite group and let ψ(G) denote the sum of element orders in G. In 2009, H.Amiri, S.M.Jafarian Amiri and I.M.Isaacs proved that if |G| = n and G is non-cyclic, then ψ(G) < ψ(C n ), where C n denotes the cyclic group of order n. In 2018 we proved that if G is non-cyclic group of order n, then ψ(G) ≤ 7 11 ψ(C n ) and equality holds if In this paper we proved that equality holds if and only if n and G are as indicated above. Moreover we proved the following generalization of this result: Theorem 4. Let q be a prime and let G be a non-cyclic group of order n, with q being the least prime divisor of n. Then ψ(G) ≤ ((q 2 -1)q+1)(q+1) q 5 +1 ψ(C n ), with equality if and only if n = q 2 k with (k, q) = 1 and G = (C q × C q ) × C k . Notice that if q = 2, then ((q 2 -1)q+1)(q+1) q 5 +1 = 7 11 .
Annali di Matematica Pura ed Applicata, Dec 1, 1986
Let 5~ be the class of groups de]i/ned by : G e J~ if and only if H ~ ~ is finitely generated whe... more Let 5~ be the class of groups de]i/ned by : G e J~ if and only if H ~ ~ is finitely generated whenever H and K are finitely generated subgroups of G. In tMs paper we get the following theorems: -Let G be a nilpotent by abelian group satisfying the maximal condition on normal subgroups. Then the ]ollowing statements are equivalent: (2) every finitely generated subgroup o/ G is polyeyelie or of finite index in G; (3) G ~ is polycyclie and G/G" e J~.
Denote by G a finite group and by ψ(G) the sum of element orders in G. If t is a positive integer... more Denote by G a finite group and by ψ(G) the sum of element orders in G. If t is a positive integer, denote by C t the cyclic group of order t and write ψ(t) = ψ(C t ). In this paper we proved the following Theorem A: Let G be a non-cyclic group of odd order n = qm, where q is the smallest prime divisor of n and (m, q) = 1. Then the following statements hold. (1) If q = 3, then 301 , and equality holds if and only if n = 3 , where p is the smallest prime bigger than q and equality holds if and only if n = qp 2 m 1 with (m 1 , p!) = 1 and G = C q × C p × C p × C m 1 .
Journal of Algebra and Its Applications, Feb 17, 2019
In this note we present a new subgroup embedding property that can be considered as an analogue o... more In this note we present a new subgroup embedding property that can be considered as an analogue of pronormality in the scope of permutability and Sylow permutability in finite groups. We prove that finite PST-groups, or groups in which Sylow permutability is a transitive relation, can be characterised in terms of this property, in a similar way as T-groups, or groups in which normality is transitive, can be characterised in terms of pronormality.
Journal of the Australian Mathematical Society, Aug 1, 1992
Groups in which every infinite set of subgroups contains a pair that permute were studied by M. C... more Groups in which every infinite set of subgroups contains a pair that permute were studied by M. Curzio, J. Lennox, A. Rhemtulla and J. Wiegold. The question whether periodic groups in this class were locally finite was left open. Here we show that if the generators of such a group G are periodic then G is locally finite. This enables us to get the following characterisation. A finitely generated group G is centre-by-finite if and only if every infinite set of subgroups of G contains a pair that permute.
Following J.S. Rose, a subgroup H of a group G is said contranormal in G if G = H G . In a certai... more Following J.S. Rose, a subgroup H of a group G is said contranormal in G if G = H G . In a certain sense, contranormal subgroups are antipodes to subnormal subgroups. It is well known that a finite group is nilpotent if and only if it has no proper contranormal subgroups. We prove that a nilpotent-by-finite group with no proper contranormal subgroup is nilpotent. There are locally nilpotent groups with a proper contranormal subgroup. We study the structure of hypercentral groups with a finite proper contranormal subgroup.
Proceedings of the American Mathematical Society, Oct 25, 1999
of subgroups such that for each ordinal α, Gα G, and G α+1 /Gα has no nonabelian free subsemigrou... more of subgroups such that for each ordinal α, Gα G, and G α+1 /Gα has no nonabelian free subsemigroup, then G is right orderable if and only if it is locally indicable. In particular if G is a radical-by-periodic group, then it is right orderable if and only if it is locally indicable.
In this paper we shall deal with periodic groups, in which each element has a prime power order. ... more In this paper we shall deal with periodic groups, in which each element has a prime power order. A group G will be called a BCP -group if each element of G has a prime power order and for each p ∈ π(G) there exists a positive integer u p such that each p-element of G is of order p i ≤ p u p . A group G will be called a BSP -group if each element of G has a prime power order and for each p ∈ π(G) there exists a positive integer v p such that each finite p-subgroup of G is of order p j ≤ p v p . Here π(G) denotes the set of all primes dividing the order of some element of G. Our main results are the following four theorems. Theorem 1: Let G be a finitely generated BCP -group. Then G has only a finite number of normal subgroups of finite index. Theorem 4: Let G be a locally graded BCP -group. Then G is a locally finite group. Theorem 7: Let G be a locally graded BSP -group. Then G is a finite group. Theorem 9: Let G be a BSP -group satisfying 2 ∈ π(G). Then G is a locally finite group.
We study an inverse problem of small doubling type. We investigate the structure of a finitely ge... more We study an inverse problem of small doubling type. We investigate the structure of a finitely generated group $G$ such that, for any set $S$ of generators of $G$ of minimal order, we have $S^2 leq 3|S|-beta$, where $beta in {1, 2, 3}$
In this note, we present a new subgroup embedding property that can be considered as an analogue ... more In this note, we present a new subgroup embedding property that can be considered as an analogue of pronormality in the scope of permutability and Sylow permutability in finite groups. We prove that finite PST-groups, or groups in which Sylow permutability is a transitive relation, can be characterized in terms of this property, in a similar way as T-groups, or groups in which normality is transitive, can be characterized in terms of pronormality.
Suppose that 12 is a variety of groups defined by means of a law v(x•,... ,x,•) = 1, and say that... more Suppose that 12 is a variety of groups defined by means of a law v(x•,... ,x,•) = 1, and say that a group G belongs to the class 12' if for any infinite subsets X•,... ,X,• there exist elements xi • Xi, for all i • {1,... ,n}, such that V(Xl,... ,x,•)= 1. Obviously the class 12 and the
Journal of the Australian Mathematical Society, 1992
Groups in which every infinite set of subgroups contains a pair that permute were studied by M. C... more Groups in which every infinite set of subgroups contains a pair that permute were studied by M. Curzio, J. Lennox, A. Rhemtulla and J. Wiegold. The question whether periodic groups in this class were locally finite was left open. Here we show that if the generators of such a group G are periodic then G is locally finite. This enables us to get the following characterisation. A finitely generated group G is centre-by-finite if and only if every infinite set of subgroups of G contains a pair that permute.
... Since the g i i are free generators for K modulo Z, we have KH KZ; also K/ KZ is free abelian... more ... Since the g i i are free generators for K modulo Z, we have KH KZ; also K/ KZ is free abelian. But ZCG HCG K , since centralizers are isolated, and so ZKZ K. The result follows. Patrizia Longobardi, Mercede Maj and Howard Smith 292 Page 3. ...
ABSTRACT Let D i be the class of groups with i conjugacy classes of infinite size and let D be th... more ABSTRACT Let D i be the class of groups with i conjugacy classes of infinite size and let D be the union of all D i for all i=0,1,2,⋯. In these terms D 0 is exactly the class of FC-groups. In this article the authors investigate the D-groups. They obtain some interesting properties of these groups, characterize periodic locally graded non-FC-groups, and study in detail D 1 -groups.
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