§ 226. Noncyclic p-groups containing only one proper normal subgroup of a given order

2016 ◽  
Keyword(s):  
Normal Subgroup ◽  
Proper Normal Subgroup

scholarly journals Elements Generating a Proper Normal Subgroup of the Cremona Group

2020 ◽  
Author(s):  
Serge Cantat ◽  
Vincent Guirardel ◽  
Anne Lonjou
Keyword(s):  
Normal Subgroup ◽  
Projective Plane ◽  
Infinite Order ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Cremona Group ◽  
Birational Transformations ◽  
Proper Normal Subgroup

Abstract Consider an algebraically closed field ${\textbf{k}}$, and let $\textsf{Cr}_2({\textbf{k}})$ be the Cremona group of all birational transformations of the projective plane over ${\textbf{k}}$. We characterize infinite order elements $g\in \textsf{Cr}_2({\textbf{k}})$ having a power $g^n$, $n\neq 0$, generating a proper normal subgroup of $\textsf{Cr}_2({\textbf{k}})$.

On non-strictly simple groups

1963 ◽  
Vol 59 (3) ◽  
pp. 531-553 ◽  
Author(s):  
P. Hall
Keyword(s):  
Normal Subgroup ◽  
Ordinal Number ◽  
Simple Groups ◽  
Proper Normal Subgroup

Ascending series. LetGbe a group and λ an ordinal number. An ascending series ofGoftypeλ is a set of subgroupsGαofG, defined for all α ≤ λ, and such that (i)G0= 1,Gλ=G; (ii)Gαis a proper normal subgroup ofGα+1for all α < λ (iii)for all limit ordinals μ ≤ λ.

scholarly journals On the asymptotic enumeration of Cayley graphs

2021 ◽  
Author(s):  
Joy Morris ◽  
Mariapia Moscatiello ◽  
Pablo Spiga
Keyword(s):  
Automorphism Group ◽  
Normal Subgroup ◽  
Regular Representation ◽  
Cayley Graphs ◽  
Asymptotic Enumeration ◽  
Undirected Graphs ◽  
Regular Group ◽  
Cayley Digraph ◽  
Graphical Regular Representation ◽  
Proper Normal Subgroup

AbstractIn this paper, we are interested in the asymptotic enumeration of Cayley graphs. It has previously been shown that almost every Cayley digraph has the smallest possible automorphism group: that is, it is a digraphical regular representation (DRR). In this paper, we approach the corresponding question for undirected Cayley graphs. The situation is complicated by the fact that there are two infinite families of groups that do not admit any graphical regular representation (GRR). The strategy for digraphs involved analysing separately the cases where the regular group R has a nontrivial proper normal subgroup N with the property that the automorphism group of the digraph fixes each N-coset setwise, and the cases where it does not. In this paper, we deal with undirected graphs in the case where the regular group has such a nontrivial proper normal subgroup.

Generalized Frattini Subgroups of Finite Groups. II

1969 ◽  
Vol 21 ◽  
pp. 418-429 ◽  
Author(s):  
James C. Beidleman
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Proper Subgroup ◽  
Subnormal Subgroup ◽  
Normal Subgroups ◽  
Frattini Subgroup ◽  
A Finite Group ◽  
Proper Normal Subgroup ◽  
Equivalent Conditions

The theory of generalized Frattini subgroups of a finite group is continued in this paper. Several equivalent conditions are given for a proper normal subgroup H of a finite group G to be a generalized Frattini subgroup of G. One such condition on H is that K is nilpotent for each normal subgroup K of G such that K/H is nilpotent. From this result, it follows that the weakly hyper-central normal subgroups of a finite non-nilpotent group G are generalized Frattini subgroups of G.Let H be a generalized Frattini subgroup of G and let K be a subnormal subgroup of G which properly contains H. Then H is a generalized Frattini subgroup of K.Let ϕ(G) be the Frattini subgroup of G. Suppose that G/ϕ(G) is nonnilpotent, but every proper subgroup of G/ϕ(G) is nilpotent. Then ϕ(G) is the unique maximal generalized Frattini subgroup of G.

scholarly journals On Normal Subgroups of Products of Nilpotent Groups

1988 ◽  
Vol 44 (2) ◽  
pp. 275-286 ◽  
Author(s):  
Bernhard Amberg ◽  
Silvana Franciosi ◽  
Francesco De Giovanni
Keyword(s):  
Normal Subgroup ◽  
Nilpotent Groups ◽  
Normal Subgroups ◽  
Proper Normal Subgroup

AbstractLet G be a group factorized by finitely many pairwise permutable nilpotent subgroups. The aim of this paper is to find conditions under which at least one of the factors is contained in a proper normal subgroup of G.

scholarly journals A Theorem on Factorizable Groups of Odd Order

1962 ◽  
Vol 20 ◽  
pp. 201-203
Author(s):  
Osamu Nagai
Keyword(s):  
Normal Subgroup ◽  
Maximal Subgroup ◽  
Factorizable Group ◽  
Odd Order ◽  
Proper Normal Subgroup

Recently, W. Feit [2] obtained some results on factorizable groups of odd order. By using his procedure and applying the theory of R. Brauer [1], we can prove the following theorem similar to that of W. Feit [2]: Theorem. Let G be a factorizable group of odd order such that G = HM where H is a subgroup of order 3p, p being a prime greater than 3, and M is a maximal subgroup of G. Then G contains a proper normal subgroup which is contained either in H or in M.

scholarly journals Groups of homeomorphisms and normal subgroups of the group of permutations

1991 ◽  
Vol 14 (3) ◽  
pp. 475-480 ◽  
Author(s):  
P. T. Ramachandran
Keyword(s):  
Normal Subgroup ◽  
Normal Subgroups ◽  
Group Of Homeomorphisms ◽  
Proper Normal Subgroup

In this paper, it is proved that no nontrivial proper normal subgroup of the group of permutations of a setXcan be the group of homeomorphisms of(X,T)for any topologyTonX.

scholarly journals A nonabelian Frobenius–Wielandt complement

1988 ◽  
Vol 31 (1) ◽  
pp. 67-69 ◽  
Author(s):  
Alberto Espuelas
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Proper Subgroup ◽  
A Finite Group ◽  
Proper Normal Subgroup

We recall the following definition (see [1]):A finite group G is said to be a Frobenius–Wielandt group provided that there exists a proper subgroup H of G and a proper normal subgroup N of H such that H∩Hg≦N if g∈G–H. Then H/N is said to be the complement of (G, H, N) (see [1] for more details and notation).

Canonical Submersion and Lie Homomorphisms Normal Subgroup

2020 ◽  
Vol 23 (1) ◽  
pp. 97-101
Author(s):  
Mikhail Petrichenko ◽  
Dmitry W. Serow
Keyword(s):  
Normal Subgroup ◽  
Canonical System ◽  
System Of Equations ◽  
Core Group ◽  
The Core ◽  
Switch Group

Normal subgroup module f (module over the ring F = [ f ] 1; 2-diffeomorphisms) coincides with the kernel Ker Lf derivations along the field. The core consists of the trivial homomorphism (integrals of the system v = x = f (t; x )) and bundles with zero switch group Lf , obtained from the condition ᐁ( ω × f ) = 0. There is the analog of the Liouville for trivial immersion. In this case, the core group Lf derivations along the field replenished elements V ( z ), such that ᐁz = ω × f. Hence, the core group Lf updated elements helicoid (spiral) bundles, in particular, such that f = ᐁU. System as an example Crocco shown that the canonical system does not permit the trivial embedding: the canonical system of equations are the closure of the class of systems that permit a submersion.

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