Some properties of n-capable and n-perfect groups (original) (raw)
Some properties of ܖ -capable and ܖ -perfect groups
2013
In this article we introduce the notion of n-capable groups. It is shown that every group G admits a uniquely determined subgroup (〖Z^n)〗^* (G) which is a characteristic subgroup and lies in the n-centre subgroup of the group G. This is the smallest subgroup of G whose factor group is n-capable. Moreover, some properties of n-central extension will be studied.
2020
Following P. Hall a soluble group whose Sylow subgroups are all abelian is called A-group. The purpose of this article is to give a new and shorter proof for a criterion on the capability of A-groups of order p^2q, where p and q are distinct primes. Subsequently we give a sufficient condition for n-capability of groups having the property that their center and derived subgroups have trivial intersection, like the groups with trivial Frattini subgroup and A-groups. An interesting necessary and sufficient condition for capability of the A-groups of square free order will be also given.
Indagationes Mathematicae, 1997
Let F, be a free group on a countable set {XI, x2, .} and V be a variety of groups, defined by the set of outer commutators V, in the free generators xi's, The paper is devoted to give the complete structure of a V-covering of V-perfect groups. Furthermore necessary and sufficient conditions for the universality of a U-central extension by a group and its V-covering group will be presented.
Journal of Group Theory, 2007
A finite group G is said to be a PST-group if, for subgroups H and K of G with H Sylow-permutable in K and K Sylow-permutable in G, it is always the case that H is Sylowpermutable in G. A group G is a T *-group if, for subgroups H and K of G with H normal in K and K normal in G, it is always the case that H is Sylow-permutable in G. In this paper, we show that finite PST-groups and finite T *-groups are one and the same. A new characterisation of soluble PST-groups is also presented.
ON THE FINITENESS PROPERTIES OF GROUPS
For an automorphism ϕ of the group G, the connection between the centralizer C G (ϕ) and the commutator [G, ϕ] is investigated and as a consequence of the Schur theorem it is shown that if G/C G (ϕ) and G ′ are both finite, then so is [G, ϕ].
Some permutability properties related to -hypercentrally embedded subgroups of finite groups
Journal of Algebra, 2003
Let F denote a saturated formation. In this paper we study some properties of F-hypercentrally embedded subgroups, i.e., those subgroups T of a finite group G such that every chief factor of G between its core and its normal closure is F-central in G. We prove that these subgroups form a sublattice of the lattice of all subgroups of G, if F is subgroup-closed. The main result of the paper is the following: if F contains the class of nilpotent groups and G is a soluble group, a subgroup T which permutes with all Sylow subgroups of G is F-hypercentrally embedded in G if and only if T permutes with some F-normalizer of G.
On finite factorizable groups*1
Journal of Algebra, 1984
ON FINITE FACTORIZABLE GROUPS 523 (I) A, with r > 5 a prime and A N A,-, . (II) M,, and either A is solvable or A N M,,. (III) M,, and either B is Frobenius of order 11 . 23 or B is cyclic of order 23 and A N M,, .
1 a Characterization of Normal Subgroups via N-Closed Sets
2016
Let (G, *) be a semigroup, D ⊆ G, and n ≥ 2 be an integer. We say that (D, *) is an n-closed subset of G if a 1 * • • • * an ∈ D for every a 1 , ..., an ∈ D. Hence every closed set is a 2-closed set. The concept of n-closed sets arise in so many natural examples. For example, let D be the set of all odd integers, then (D, +) is a 3-closed subset of (Z, +) that is not a 2-closed subset of (Z, +). If K = {1, 4, 7, 10, ...} , then (K, +) is a 4-closed subset of (Z, +) that is not an n-closed subset of (Z, +) for n = 2, 3. In this paper, we show that if (H, *) is a subgroup of a group (G, *) such that [H : G] = n < ∞, then H is a normal subgroup of G if and only if every left coset of H is an n + 1-closed subset of G.
On a generalization of M-group
arXiv preprint arXiv:1206.5067, 2012
Abstract: In this paper, we will show that if for every nonlinear complex irreducible character of a finite group G, some multiple of it is induced from an irreducible character of some proper subgroup of G, then G is solvable. This is a generalization of Taketa's Theorem on the solvability of M-group.
On certain minimal non-Y-groups for some classes Y
2015
Let {θn} ∞ n=1 be a sequence of words. If there exists a positive integer n such that θm(G) = 1 for every m ≥ n , then we say that G satisfies (*) and denote the class of all groups satisfying (*) by X {θn} ∞ n=1. If for every proper subgroup K of G , K ∈ X {θn} ∞ n=1 but G / ∈ X {θn} ∞ n=1 , then we call G a minimal non-X {θn} ∞ n=1-group. Assume that G is an infinite locally finite group with trivial center and θi(G) = G for all i ≥ 1. In this case we mainly prove that there exists a positive integer t such that for every proper normal subgroup N of G , either θt(N) = 1 or θt(CG(N)) = 1. We also give certain useful applications of the main result.
Journal of Algebra, 1980
A well-known theorem of Wielandt states that a finite group G is nilpotent if and only if every maximal subgroup of G is normal in G. The structure of a nonnilpotent group, each of whose proper subgroups is nilpotent, has been analyzed by Schmidt and R6dei [5, Satz 5.1 and Satz 5.2, pp. 280-281]. In [1], Buckley investigated the structure of a PN-group (i.e., a finite group in which every minimal subgroup is normal), and proved (i) that a PN-group of odd order is supersolvable, and (ii) that certain factor groups of a PN-group of odd prime power order are also PN-groups. Earlier, Gaschiitz and It5 [5, Satz 5.7, p. 436] had proved that the commutator subgroup of a finite PN-group is p-nilpotent for each odd prime p. This paper is a sequel to [9] and our object here is to prove the following statement. THEOREM. If G is a finite nonPN-group, each of whose proper subgroups is a PN-group, then one of the following statements is true: (a) G is the dihedral group of order 8.
Characterizations of some groups in terms of centralizers
2021
A group G is said to be n-centralizer if its number of element centralizers | Cent(G) |= n, an F-group if every non-central element centralizer contains no other element centralizer and a CA-group if all non-central element centralizers are abelian. For any n > 11, we prove that if G is an n-centralizer group, then | G Z(G) |≤ 2(n − 4) (n−4) 2 , which improves an earlier result (it is known that | G Z(G) |≤ (n − 2) for n ≤ 11). We also prove that if G is an arbitrary n-centralizer F-group, then gcd(n − 2, | G Z(G) |) 6= 1. For a finite F-group G, we show that | Cent(G) |≥ |G| 2 iff G ∼= A4, D2n (where n is odd) or an extraspecial 2-group. Among other results, for a finite group G with non-trivial center, it is proved that | Cent(G) |= |G| 2 iff G is an extraspecial 2-group. We give a family of F-groups which are not CA-groups and extend an earlier result.
Atti della Accademia Peloritana dei Pericolanti : Classe di Scienze Fisiche, Matematiche e Naturali, 2018
In this article we introduce the notion of e-group as a new generalization of a group. The condition for a group to be an e-group is given. The characterization of some properties is established and some results follow. * 2 a b c d a a a a a b a b c d c c c a d d b d b c Then (G; * 1 , A) satisfies (G1) and (eG2) but does not satisfy (eG3). In addition, (G; * 2 ; A) satisfies (eG2) and (eG3) but does not satisfy (G1), since
The influence of C- Z-permutable subgroups on the structure of finite groups
2018
Let Z be a complete set of Sylow subgroups of a finite group G, that is, for each prime p dividing the order of G, Z contains exactly one and only one Sylow p-subgroup of G, say Gp. Let C be a nonempty subset of G. A subgroup H of G is said to be C-Z-permutable (conjugateZ-permutable) subgroup of G if there exists some x ∈ C such that HGp = GpH, for all Gp ∈ Z. We investigate the structure of the finite group G under the assumption that certain subgroups of prime power orders of G are C-Z-permutable subgroups of G. 2010 Mathematics Subject Classifications: 20D10, 20D15, 20D20, 20F16.
On the autocentralizer subgroups of finite –groups
TURKISH JOURNAL OF MATHEMATICS, 2020
Let G be a finite group and Aut(G) be the group of automorphisms of G. Then, the autocentralizer of an automorphism α ∈ Aut(G) in G is defined as CG(α) = {g ∈ G|α(g) = g}. Let Acent(G) = {CG(α)|α ∈ Aut(G)}. If |Acent(G)| = n, then G is an n-autocentralizer group. In this paper, we classify all n-autocentralizer abelian groups for n = 6, 7 and 8. We also obtain a lower bound on the number of autocentralizer subgroups for p-groups, where p is a prime number. We show that if p ̸ = 2, there is no n-autocentralizer p-group for n = 6, 7. Moreover, if p = 2, then there is no 6-autocentralizer p-group.
On Groups Which Contain No HNN-Extensions
International Journal of Algebra and Computation, 2007
A group is called HNN-free if it has no subgroups that are nontrivial HNN-extensions. We prove that finitely generated HNN-free implies virtually polycyclic for a large class of groups. We also consider finitely generated groups with no free subsemigroups of rank 2 and show that in many situations such groups are virtually nilpotent. Finally, as an application of our results, we determine the structure of locally graded groups in which every subgroup is pronormal, thus generalizing a theorem of Kuzennyi and Subbotin.