Induction and Restriction of Lifts of π-Partial Characters

Author(s):

Mark L. Lewis

Keyword(s):

Normal Subgroups ◽

Lattices Of Subgroups ◽

In this paper, we study lattices of subgroups that can be used to obtain lifts of π-partial characters. In particular, we find a condition on the lattices so that the associated lifts will be well-behaved regarding induction and restriction on normal subgroups. We show that the lattices obtained from all subnormal subgroups have this property, but the lattices obtained from all normal subgroups do not.

A NOTE ON GROUPS WHOSE PROPER LARGE SUBGROUPS HAVE A TRANSITIVE NORMALITY RELATION

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972716000848 ◽

2016 ◽

Vol 95 (1) ◽

pp. 38-47 ◽

Author(s):

FRANCESCO DE GIOVANNI ◽

MARCO TROMBETTI

Keyword(s):

Soluble Group ◽

Subnormal Subgroup ◽

Normal Subgroups ◽

Transitive Relation ◽

Formula Group ◽

A group $G$ is said to have the $T$-property (or to be a $T$-group) if all its subnormal subgroups are normal, that is, if normality in $G$ is a transitive relation. The aim of this paper is to investigate the behaviour of uncountable groups of cardinality $\aleph$ whose proper subgroups of cardinality $\aleph$ have a transitive normality relation. It is proved that such a group $G$ is a $T$-group (and all its subgroups have the same property) provided that $G$ has an ascending subnormal series with abelian factors. Moreover, it is shown that if $G$ is an uncountable soluble group of cardinality $\aleph$ whose proper normal subgroups of cardinality $\aleph$ have the $T$-property, then every subnormal subgroup of $G$ has only finitely many conjugates.

ON GENERALISED PRONORMAL SUBGROUPS OF FINITE GROUPS

Glasgow Mathematical Journal ◽

10.1017/s0017089514000159 ◽

2014 ◽

Vol 56 (3) ◽

pp. 691-703 ◽

Author(s):

A. BALLESTER-BOLINCHES ◽

J. C. BEIDLEMAN ◽

A. D. FELDMAN ◽

M. F. RAGLAND

Keyword(s):

Finite Group ◽

Finite Groups ◽

Subnormal Subgroup ◽

Normal Subgroups ◽

Transitive Relation ◽

Formula Group ◽

Permutable Subgroups ◽

Sylow Permutable ◽

A Finite Group ◽

AbstractFor a formation $\mathfrak F$, a subgroup M of a finite group G is said to be $\mathfrak F$-pronormal in G if for each g ∈ G, there exists x ∈ 〈U,Ug〉$\mathfrak F$ such that Ux = Ug. Let f be a subgroup embedding functor such that f(G) contains the set of normal subgroups of G and is contained in the set of Sylow-permutable subgroups of G for every finite group G. Given such an f, let fT denote the class of finite groups in which f(G) is the set of subnormal subgroups of G; this is the class of all finite groups G in which to be in f(G) is a transitive relation in G. A subgroup M of a finite group G is said to be $\mathfrak F$-normal in G if G/CoreG(M) belongs to $\mathfrak F$. A subgroup U of a finite group G is called K-$\mathfrak F$-subnormal in G if either U = G or there exist subgroups U = U0 ≤ U1 ≤ . . . ≤ Un = G such that Ui–1 is either normal or $\mathfrak F$-normal in Ui, for i = 1,2, …, n. We call a finite group G an $fT_{\mathfrak F}$-group if every K-$\mathfrak F$-subnormal subgroup of G is in f(G). In this paper, we analyse for certain formations $\mathfrak F$ the structure of $fT_{\mathfrak F}$-groups. We pay special attention to the $\mathfrak F$-pronormal subgroups in this analysis.

On R0-Closed Classes, and Finitely Generated Groups

Canadian Journal of Mathematics ◽

10.4153/cjm-1970-021-x ◽

1970 ◽

Vol 22 (1) ◽

pp. 176-184 ◽

Author(s):

Rex Dark ◽

Akbar H. Rhemtulla

Keyword(s):

Normal Subgroups ◽

Finitely Generated ◽

Maximal Condition ◽

Soluble Groups ◽

Finitely Generated Group ◽

Finitely Generated Groups ◽

Central Factors ◽

1.1. If a group satisfies the maximal condition for normal subgroups, then all its central factors are necessarily finitely generated. In [2], Hall asked whether there exist finitely generated soluble groups which do not satisfy the maximal condition for normal subgroups but all of whose central factors are finitely generated. We shall answer this question in the affirmative. We shall also construct a finitely generated group all of whose subnormal subgroups are perfect (and which therefore has no non-trivial central factors), but which does not satisfy the maximal condition for normal subgroups. Related to these examples is the question of which classes of finitely generated groups satisfy the maximal condition for normal subgroups. A characterization of such classes has been obtained by Hall, and we shall include his result as our first theorem.

Groups whose lattices of normal subgroups are distributive

Glasgow Mathematical Journal ◽

10.1017/s0017089500007710 ◽

1989 ◽

Vol 31 (2) ◽

pp. 183-188 ◽

Author(s):

Rolf Brandl

Keyword(s):

Finite Groups ◽

Basic Result ◽

Distributive Lattices ◽

Normal Subgroups ◽

Various authors deal with distributive sublattices of the lattice ℒ(G) of subgroups of a group G. Perhaps the most basic result in this direction is due to O. Ore [9]: ℒ(G) is distributive if and only if G is locally cyclic.In [11] and [12] finite groups with distributive lattices of subnormal subgroups were considered, while [3], [4], [7], [8], [10] and [13] deal with the case of groups G whose lattice N(G) of normal subgroups is distributive. Such groups were called DLN-groups in [10].

On certain subgroups of a join of subnormal subgroups

Glasgow Mathematical Journal ◽

10.1017/s0017089500005486 ◽

1984 ◽

Vol 25 (1) ◽

pp. 103-105 ◽

Cited By ~ 4

Author(s):

Howard Smith

Keyword(s):

Finite Index ◽

Finite Groups ◽

Normal Subgroups ◽

Normal Core ◽

1. Introduction: Suppose the group G is generated by subnormal subgroups H and K, and that A, B are normal subgroups of finite index in H, Krespectively. The following question has been asked by J. C. Lennox: Under what circumstances is the subgroupJ = (A, B) subnormal in G? In particular, it is of interest to know when J has finite index in G, for, if this is the case, we may factor out by the normal core of J in G and apply Wielandt's theorem on joins of subnormal subgroups of finite groups [11] to deduce that J is subnormal in G. Here we prove the following result.

On a class of finite solvable groups

Open Mathematics ◽

10.2478/s11533-013-0264-2 ◽

2013 ◽

Vol 11 (9) ◽

Author(s):

James Beidleman ◽

Hermann Heineken ◽

Jack Schmidt

Keyword(s):

Solvable Group ◽

Prime Order ◽

Solvable Groups ◽

Finite Solvable Group ◽

Normal Subgroups ◽

Nilpotent Residual ◽

Quotient Groups ◽

AbstractA finite solvable group G is called an X-group if the subnormal subgroups of G permute with all the system normalizers of G. It is our purpose here to determine some of the properties of X-groups. Subgroups and quotient groups of X-groups are X-groups. Let M and N be normal subgroups of a group G of relatively prime order. If G/M and G/N are X-groups, then G is also an X-group. Let the nilpotent residual L of G be abelian. Then G is an X-group if and only if G acts by conjugation on L as a group of power automorphisms.

On Groups with Chain Conditions

Canadian Journal of Mathematics ◽

10.4153/cjm-1971-015-8 ◽

1971 ◽

Vol 23 (1) ◽

pp. 151-159

Author(s):

Bernhard Amberg

Keyword(s):

Minimum Condition ◽

Normal Subgroups ◽

Maximum Condition ◽

Finiteness Conditions ◽

Chain Conditions ◽

Radical Group ◽

Abelian Subgroups ◽

Maximum Minimum ◽

Our aim in this note is to generalize results of Baer in [3; 5]. In § 1 an arbitrary formation n is considered, the key result being Proposition 1.5. This is applied in § 2 to characterize various finiteness conditions, for example the classes of groups with maximum [minimum] condition on subgroups, subnormal subgroups, and normal subgroups respectively, or the class of (not necessarily soluble) polyminimax groups (see Theorems 2.1 and 2.6). These results may also be regarded as generalizations of the well-known theorem of Malcev-Baer that a radical group satisfies the maximum condition [is a polyminimax group] if all its abelian subgroups satisfy the maximum condition [are minimax groups].

Nuclei and Lifts of π-Partial Characters

Algebra Colloquium ◽

10.1142/s1005386709000182 ◽

2009 ◽

Vol 16 (01) ◽

pp. 167-180 ◽

Author(s):

Mark L. Lewis

Keyword(s):

Normal Subgroups ◽

Common Generalization ◽

In this paper, we obtain a common generalization of the Isaacs and Navarro constructions of nuclei for characters of π-separable groups. The construction of Isaacs used the lattice of all subnormal subgroups, and the construction of Navarro used the lattice of all normal subgroups. We show that a generalized nucleus can be constructed using other lattices of subnormal subgroups, and these nuclei have properties similar to the properties of the nuclei constructed by Isaacs and Navarro.