Groups whose maximal cyclic subgroups are independent by Louis Weisner

Cover of: Groups whose maximal cyclic subgroups are independent | Louis Weisner

Published in New York .

Written in English

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  • Group theory.

Edition Notes

Book details

Statementby Louis Weisner.
LC ClassificationsQA171 .W4
The Physical Object
Pagination17 p. ;
Number of Pages17
ID Numbers
Open LibraryOL6654107M
LC Control Number23007882

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We consider the problem of classifying those groups whose maximal cyclic subgroups are maximal. We give a complete classification of those groups with Cited by: 1.

Free shipping for non-business customers when ordering books at De Gruyter Online. Please find details to our shipping fees here. RRP: Recommended Retail Price. Print Flyer; Overview; Content; Book Book Series.

Previous chapter. Next chapter. Appendix 2-groups whose maximal cyclic subgroups of order > 2 are self-centralizing Berkovich. Let G be a solvable non-cyclic group having cyclic 2-maximal subgroups.

Then every proper subgroup of G is a cyclic group or a minimal non-cyclic group, and each minimal non-cyclic group is maximal in by: 1.

We consider the problem of classifying those groups whose maximal cyclic subgroups are maximal. We give a complete classification of those groups with this property and which are either soluble or.

abelian p-groups, whose structure is well-known. In Section 3 we deal with infinite groups in Σ. We completely de- Then the Sylow p-subgroups of G are cyclic and, for primes r 6= p, the Sylow r-subgroups are abelian.

cyclic, there are maximal subgroups P1 and P2 of P with P1 6= P2. It follows that X normalizes both P1R and P2R. This definition does not depend on the choice of the presentation of G.

In his book ([8], p) claimed that if G1 and G2 are torsion-free hyperbolic and U and V are maximal cyclic subgroups in G1 and G2 respectively, then the amalgamated free product G1 ∗U=V G2 is also hyperbolic. The following notations will be used for the free product G of two groups A, B with the subgroup Ugenerated by ux, u2, in A and the subgroup V generated by t>u v2, in B amalgamated under an isomorphism File Size: 2MB.

I am having difficulty seeing if it is the case that if all subgroups of a group are cyclic, that the group itself is cyclic. Stack Exchange Network Stack Exchange network consists of Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Moreover, the structure of finite non-T-groups whose maximal subgroups of even order are solvable T-groups may be difficult to give if we do not use normality. View Show abstractAuthor: Marcel Herzog. subgroup rank of a group. groups with same order and subgroup rank of a group | groups with same prime-base logarithm of order and subgroup rank of a group | groups with same subgroup rank of Groups whose maximal cyclic subgroups are independent book group.

All proper subgroups are cyclic, dihedral, semidihedral or Klein four-groups. Consider each of the Z (pi)n as a cyclic group of order pnii in multiplicative notation.

Let m be the lcm of all the pnii for i = 1, 2,r. Clearly m ≤ p1n1p2n2⋯prnr. If ai ∈ Z (pi)n then (ai) (pin) = 1 and hence ami = 1. Therefore for all α ∈ G, we have αm = 1; that is, every element of G is a root of xm = 1.

Book chapterFull text access. CHAPTER II - DIRECT SUM OF CYCLIC GROUPS Pages This chapter and the next one are devoted to the study of direct sums of cyclic groups, resp.

of quasicyclic and full rational groups. Their importance stems from the fact that their structure is completely known and easily described. JOURNAL OF ALGEBRA 1, () Groups whose maximal cyclic subgroups are independent book the Maximal Subgroups of Finite Simple Groups* DANIEL GORENSTEIN dark University, Worcester, Massachusetts AND JOHN H.

WALTER University of Illinois, Urbana, Illinois Communicated by B. Huppert Received Janu I. INTRODUCTION In the course of their proof of the solvability of groups of odd order, W. Feit and J. Thompson [7] Cited by: Moreover, we classify all ï¬ nite non-cyclic abelian groups whose intersection graphs of cyclic subgroups are planar.

Also for any group G, we determine the independence number, clique cover number of Ic(G) and show that Ic(G) is weakly α by: 3. Groups of Prime Power Order. Volume 3 Free shipping for non-business customers when ordering books at De Gruyter Online. Please find Contact Persons; Book Book Series. Previous chapter.

Next chapter § 2-groups all of whose nonnormal subgroups are either cyclic or of maximal class. 30,00 € / $ / £ Get Access to Full Text. maximal subgroups of G. This useful second definition of the (/«-subgroup is also due to Frattini.

As every maximal subgroup of a group of order pm, p being a prime number, is of order pm~l, it results directly from this second definition that the (fr-subgroup of any group of order pm is the cross-cut of all its subgroups of index p.

Let G be a nonabelian 2-group of exponent 4 all of whose cyclic subgroups of order 4 are normal in G. Then G is one of the following groups: (a) G has an abelian subgroup H of exponent 4 and index 2 in G and there is an involution in G − H which inverts H, i.e., which inverts each element in H.

(b)Cited by: 2. Chin. Ann. Math. 36B(1),11–30 DOI: /s Chinese Annals of Mathematics, Series B c The Editorial Office of CAM and Springer-Verlag Berlin Heidelberg 2. A group G is called solvable if it has a subnormal series whose factor groups (quotient groups) are all abelian, that is, if there are subgroups 1 = G 0 group, for j = 1, 2,k.

The maximal subgroups M constructed so far in this paper are all free products of cyclic groups of order p (= 3 for the modular group) and q (= 2 for the Hecke groups), in bijective correspondence with the fixed points of x and y on Ω.

(More generally, by the Kurosh Subgroup Theorem (see [11, Chapter IV, Theorem ]), any subgroup of Γ = C p * C q is a free product of subgroups C Author: Gareth A.

Jones. In this article we describe finite solvable groups whose 2-maximal subgroups are nilpotent (a 2-maximal subgroup of a group).

Unsolvable groups with this property were described in [2,3]. View. Finite groups with non-nilpotent maximal subgroups Lemma Let P be a Sylow p-subgroup of G, where p is the largest prime dividing the order of G. Assume that P is not normal in G, and M.

All subgroups and quotient groups of cyclic groups are cyclic. Specifically, all subgroups of Z are of the form m = m Z, with m a positive integer. All of these subgroups are distinct from each other, and apart from the trivial group {0} = 0 Z, they all are isomorphic to Z.

In fact, such groups have a maximal quotient which is a 3-group of maximal class. This paper is a part of classification of finite p-groups with a minimal non-abelian subgroup of index p, and.

This group has two nontrivial subgroups: J={0,4} and H={0,2,4,6}, where J is also a subgroup of H. The Cayley table for H is the top-left quadrant of the Cayley table for G. The group G is cyclic, and so are its subgroups. In general, subgroups of cyclic groups are also cyclic.

Example-3 (Groups of order 60): If o(G)=60 and G has more than one Sylow 5-subgroups, then G is simple. Cyclic group orders. Some numbers n are such that every group of order n is cyclic.

One can show that n = 15 is such a number using the Sylow theorems: Let G be a group of order 15 = 3 5 and n 3 be the number of. Groups with large normal closures of nonnormal cyclic subgroups Groups all of whose cyclic subgroups of composite orders are normal p-groups generated by elements of given order; Groups all of whose cyclic subgroups of composite orders are normal p-groups generated by elements of given order; The power graph Γ G of a finite group G is the graph whose vertex set is G, two distinct elements being adjacent if one is a power of the this paper, we give sharp lower and upper bounds for the independence number of Γ G and characterize the groups achieving the bounds.

Moreover, we determine the independence number of Γ G if G is cyclic, dihedral or generalized by: 4. Thanks for contributing an answer to Mathematics Stack Exchange. Please be sure to answer the question. Provide details and share your research.

But avoid Asking for help, clarification, or responding to other answers. Making statements based on opinion; back them up with references or personal experience.

Use MathJax to format equations. NON-CYCLIC GRAPH OF A GROUP 3 finite. In Section 5, we characterize finite non-cyclic groups whose non-cyclic graphs are regular. In Section 6, we characterize finite non-cyclic abelian groups whose non-cyclic graphs have exactly two kind degrees.

Section 7 contains some results on groups whose non-cyclic graphs are isomorphic. MORE RESULTS ON COMMUTATOR SUBGROUPS. INVARIANT SERIES AND CHIEF SERIES. KEY WORDS. SUMMARY. SELF ASSESMENT QUESTIONS. SUGGESTED READINGS. OBJECTIVE. Objective of this Chapter is to study some properties of groups by studying the properties of the series of its subgroups and factor groups.

Size: KB. In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group S n defined over a finite set of n symbols consists of the permutation operations that can be performed on the n symbols.

Quasidihedral group. In mathematics, the quasi-dihedral groups, also called semi-dihedral groups, are certain non-abelian groups of order a power of 2. For every positive integer n greater than or equal to 4, there are exactly four isomorphism classes of nonabelian groups of order 2 n which have a cyclic subgroup of index 2.

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order (the axiom of commutativity).Abelian groups generalize the arithmetic of addition of integers.

They are named after Niels Henrik Abel. The concept of an abelian group is one of the first concepts encountered. In mathematics, the classification of finite simple groups states that every finite simple group is cyclic, or alternating, or in one of 16 families of groups of Lie type, or one of 26 sporadic groups.

General linear group of a vector space. If V is a vector space over the field F, the general linear group of V, written GL(V) or Aut(V), is the group of all automorphisms of V, i.e.

the set of all bijective linear transformations V → V, together with functional composition as group V has finite dimension n, then GL(V) and GL(n, F) are isomorphic.

Stack Exchange network consists of Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share. In group theory, the quaternion group Q 8 (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to a certain eight-element subset of the quaternions under multiplication.

It is given by the group presentation where e is the identity element and e. This group is called the factor or quotient group of G and N. Simple group: In mathematics, a Simple group is a nontrivial group whose only normal subgroups are the trivial group itself.A group that is not simple can be broken into two smaller groups,a normal subgroup and quotient group,and the process canbe repeated.

Abstract. The cyclic graph of a finite group is as follows: take as the vertices of and join two distinct vertices and if is cyclic. In this paper, we investigate how the graph theoretical properties of affect the group theoretical propertieswe consider some properties of and characterize certain finite groups whose cyclic graphs have some by: 6.

Hirsch length. The Hirsch length or Hirsch number of a polycyclic group G is the number of infinite factors in its subnormal series. If G is a polycyclic-by-finite group, then the Hirsch length of G is the Hirsch length of a polycyclic normal subgroup H of G, where H has finite index in is independent of choice of subgroup, as all such subgroups will have the same Hirsch length.If the group is finite, it is a convenient way to visualize the lattice using its Hasse diagram, where the bottom element represents the identity subgroup 1, the top element the group itself, and between two elements of the lattice a line segment is drawn whenever the lower subgroup is a maximal File Size: and we construct an isometric embedding of a locally compact tree into the bi-invariant Cayley graph of a nonabelian free group.

We investigate the geometry of cyclic subgroups. We observe that in many classes of groups, cyclic subgroups are either bounded or detected by homogeneous quasimorphisms. We call this property.

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