On a Generalization of Hamiltonian Groups and a Dualization of PN-Groups
TL;DR: In this article, the intersection of the normalizers of all non-cyclic subgroups of a finite group G is studied and the results of Passman, Bozikov, and Janko are extended to non-nilpotent finite groups.
Abstract: Baer and Wielandt in 1934 and 1958, respectively, considered that the intersection of the normalizers of all subgroups of G and the intersection of the normalizers of all subnormal subgroups of G. In this article, for a finite group G, we define the subgroup S(G) to be intersection of the normalizers of all non-cyclic subgroups of G. Groups whose noncyclic subgroups are normal are studied in this article, as well as groups in which all noncyclic subgroups are normalized by all minimal subgroups. In particular, we extend the results of Passman, Bozikov, and Janko to non-nilpotent finite groups.
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01 Jan 2010
Abstract: 1. (i) Suppose K is a conjugacy class of Sn contained in An; then K is called split if K is a union of two conjugacy classes of An. Show that the number of split conjugacy classes contained in An is equal to the number of characters χ ∈ Irr(Sn) such that χAn is not irreducible. (Hint. Consider the vector space of class functions on An which are invariant under conjugation by the transposition (12).)
613 citations
15 Nov 2016
TL;DR: In this article, the authors specify all the known findings related to the norms of the group and their generalizations, and special attention is paid to the analysis of their own study of different generalized norms, particularly the norm of non-cyclic subgroups.
Abstract: In this survey paper the authors specify all the known findings related to the norms of the group and their generalizations. Special attention is paid to the analysis of their own study of different generalized norms, particularly the norm of non-cyclic subgroups, the norm of Abelian non-cyclic subgroups, the norm of infinite subgroups, the norm of infinite Abelian subgroups and the norm of other systems of Abelian subgroups.
7 citations
Cites background from "On a Generalization of Hamiltonian ..."
...Zhang [98,99] studied the properties of the norm NG of non-cyclic subgroups in the class of finite groups and its influence on the group....
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TL;DR: In this paper, the authors define the D*-norm, denoted by D*(G), to be the intersection of the normalizers of the derived subgroups of all subgroups H of G.
Abstract: Let G be a finite group. Inspired by the properties of D(G), we define the D*-norm, denoted by D*(G), to be the intersection of the normalizers of the derived subgroups of all subgroups H of G such...
4 citations
TL;DR: In this article, the authors classify two types of finite groups with certain normality conditions, namely SSN groups and groups with all noncyclic subgroups normal, which are key ingredients in the study of the multiplicative Jordan decomposition problem for group rings.
Abstract: We classify two types of finite groups with certain normality conditions, namely SSN groups and groups with all noncyclic subgroups normal. These conditions are key ingredients in the study of the multiplicative Jordan decomposition problem for group rings.
3 citations
Cites background from "On a Generalization of Hamiltonian ..."
...After our article was completed, we became aware of reference [13], written by the same authors who considered nilpotent groups having property NCN....
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References
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Book•
19 Oct 2011
TL;DR: A detailed introduction to the theory of groups: finite and infinite; commutative and non-commutative is given in this article, where the reader is provided with only a basic knowledge of modern algebra.
Abstract: This is a detailed introduction to the theory of groups: finite and infinite; commutative and non-commutative. Presupposing only a basic knowledge of modern algebra, it introduces the reader to the different branches of group theory and its principal accomplishments.
3,406 citations
"On a Generalization of Hamiltonian ..." refers background or methods in this paper
...On the other hand, by the Schur-Zassensaus theorem (Robinson [21], p. 253, Theorem 9.1.2), NG P = P M , where M is a Hall p′-subgroup of NG P and hence G = Fp G M ....
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...By a theorem of Itô (Robinson [21], p. 296, Theorem 10.3.3), K = Q R, where Q is a normal q-subgroup, exp Q = q or 4, and R is a cyclic r-subgroup, for a prime r = q....
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...As G is p-solvable, by Robinson [21], p....
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...Since CG P ≤ Fp G by Robinson ([21], p. 269, Theorem 9.3.1), M ≤ Fp G Now G = Fp G M , it follows that G = Fp G ....
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...As G is p-solvable, by Robinson [21], p. 269, Theorem 9.3.1, we know CG Op G ≤ Op G We now claim that G is q-nilpotent for any prime q = p....
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01 Jan 2010
Abstract: 1. (i) Suppose K is a conjugacy class of Sn contained in An; then K is called split if K is a union of two conjugacy classes of An. Show that the number of split conjugacy classes contained in An is equal to the number of characters χ ∈ Irr(Sn) such that χAn is not irreducible. (Hint. Consider the vector space of class functions on An which are invariant under conjugation by the transposition (12).)
613 citations
TL;DR: In this article, it was shown that if a group of minimal order belonging to p but not to g is a saturated formation, then the Fitting subgroup of G is the unique minimal normal subgroup.
Abstract: Let p be a class of finite soluble groups which is closed under epimorphic images and let g be a saturated formation. Then if G is a group of minimal order belonging to p but not to g , F(G) , the Fitting subgroup of G , is the unique minimal normal subgroup of G . It is to groups with this property that the following proposition is applicable.
238 citations
TL;DR: In this paper, it was shown that a non-abelian primitive group P in which every subgroup is abelian is always solvable, and that the theory of these groups presents remarkably few difficulties except such as are involved in abelians.
Abstract: Several years ago Dedekind and others investigated the groups in which every subgroup is invariant, and found that the theory of these groups presents remarkably few difficulties except such as are involved in abelian groups. The non-abelian groups in which every subgroup is abelian present a parallel example of simple and general results. The following are some of the most important ones : All such groups are solvable. Their orders cannot be divided by more than two distinct primes. Every commutator is of prime order. When the order ispaqß, (p and q being prime; a, /3> 0), there are just qß subgroups of order pa and there is only one subgroup of order qß . The former are cyclic and the latter is of type (1, 1, 1, •••). When the order is pa, there are just p + 1 subgroups of order p°-~l and none of them involves more than three invariants. If there are three invariants at least one of them must be of order p. Let G represent any non-abelian group in which every subgroup is abelian. We shall first prove that G is solvable. If G is represented as a transitive substitution group it will be either primitive or imprimitive. In the latter case it will be isomorphic with some primitive group P.-f The subgroup of G which corresponds to identity in P is abelian and every subgroup of P is abelian. The group G is solvable whenever P is solvable. Hence it remains to prove that a non-abelian primitive group P in which every subgroup is abelian is always solvable. Let A*, be the subgroup of P which is composed of all the substitutions omitting a given letter. Since P is non-regular,J P includes at least one substitution besides the identity. If two conjugates of Px had a common substitution besides the identity, this substitution would be invariant under P, since Px is a maximal subgroup of P. Hence P must be of class n — 1, n being the degree ofA\\ Therefore P contains an invariant subgroup of order w, § with respect to which the quotient group is simply isomorphic with Px. As
228 citations
"On a Generalization of Hamiltonian ..." refers background in this paper
...Now the theorem of Miller and Moreno [18] yields that K is solvable and hence abelian....
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