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 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 $\textsf{Cr}_2({\textbf{k}})$.

scholarly journals Simple groups of dynamical origin

2017 ◽  
Vol 39 (3) ◽  
pp. 707-732 ◽  
Author(s):  
V. NEKRASHEVYCH
Keyword(s):  
Normal Subgroup ◽  
Group Action ◽  
Simple Groups ◽  
Full Group ◽  
Normal Subgroups ◽  
Finitely Generated ◽  
Alternating Groups ◽  
Étale Groupoid ◽  
Dynamical Origin

We associate with every étale groupoid $\mathfrak{G}$ two normal subgroups $\mathsf{S}(\mathfrak{G})$ and $\mathsf{A}(\mathfrak{G})$ of the topological full group of $\mathfrak{G}$, which are analogs of the symmetric and alternating groups. We prove that if $\mathfrak{G}$ is a minimal groupoid of germs (e.g., of a group action), then $\mathsf{A}(\mathfrak{G})$ is simple and is contained in every non-trivial normal subgroup of the full group. We show that if $\mathfrak{G}$ is expansive (e.g., is the groupoid of germs of an expansive action of a group), then $\mathsf{A}(\mathfrak{G})$ is finitely generated. We also show that $\mathsf{S}(\mathfrak{G})/\mathsf{A}(\mathfrak{G})$ is a quotient of $H_{0}(\mathfrak{G},\mathbb{Z}/2\mathbb{Z})$.

scholarly journals PGL2(q) cannot be determined by its cs

Filomat ◽  
2020 ◽  
Vol 34 (5) ◽  
pp. 1713-1719
Author(s):  
Neda Ahanjideh
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Conjugacy Class ◽  
Prime Power ◽  
Simple Groups ◽  
Conjugacy Class Sizes ◽  
Direct Factor ◽  
Almost Simple Groups ◽  
Trivial Element ◽  
A Finite Group

For a finite group G, let Z(G) denote the center of G and cs*(G) be the set of non-trivial conjugacy class sizes of G. In this paper, we show that if G is a finite group such that for some odd prime power q ? 4, cs*(G) = cs*(PGL2(q)), then either G ? PGL2(q) X Z(G) or G contains a normal subgroup N and a non-trivial element t ? G such that N ? PSL2(q)X Z(G), t2 ? N and G = N. ?t?. This shows that the almost simple groups cannot be determined by their set of conjugacy class sizes (up to an abelian direct factor).

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.

Some Characterizations of a Normal Subgroup of a Group and Isotopic Classes of Transversals

2016 ◽  
Vol 23 (03) ◽  
pp. 409-422 ◽  
Author(s):  
Vipul Kakkar ◽  
R. P. Shukla
Keyword(s):  
Normal Subgroup ◽  
Finite Index ◽  
Dihedral Group ◽  
Elementary Proof ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Right Loop

Let G be a group and H be a subgroup of G which is either finite or of finite index in G. In this paper, we give some characterizations for the normality of H in G. As a consequence we get a very short and elementary proof of the main theorem of a paper of Lal and Shukla, which avoids the use of the classification of finite simple groups. Further, we study the isotopy between the transversals in some groups and determine the number of isotopy classes of transversals of a subgroup of order 2 in D2p, the dihedral group of order 2p, where p is an odd prime and the isotopism classes are formed with respect to induced right loop structures.

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