Relation Between Groups with Basis Property and Groups with Exchange Property (original) (raw)
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On Sets of PP-Generators of Finite Groups
Bulletin of the Australian Mathematical Society, 2014
The classes of finite groups with minimal sets of generators of fixed cardinalities, named B-groups, and groups with the basis property, in which every subgroup is a B-group, contain only p-groups and some {p, q}-groups. Moreover, abelian B-groups are exactly p-groups. If only generators of prime power orders are considered, then an analogue of property B is denoted by B pp and an analogue of the basis property is called the pp-basis property. These classes are larger and contain all nilpotent groups and some cyclic q-extensions of p-groups. In this paper we characterise all finite groups with the pp-basis property as products of p-groups and precisely described {p, q}-groups.
Acta Mathematica Academiae Scientiarum Hungaricae
w 5. p.basic subgroups of arbitrary abelian groups KULIKOV [8] introduced the notion of basic subgroups of abelian p-groups which has proved to be one of the most important notions in the theory of p-groups of arbitrary power. Basic subgroups can be defined in any module over the ring of p-adic integers, or, more generally, over any discrete valuation ring. Here we want to give a generalization of basic subgroups to any group so that it coincides with the old concept whenever the group is primary. In the general case, to every prime p, one can define p-basic subgroups where in the definition the prime p plays a distinguished role. The p-basic subgroups are not isomorphic for different primes, but are uniquely determined (up to isomorphism) by the group and the prime p. We shall see that p-basic subgroups are useful in certain investigations. Let G be an arbitrary (abelian) group l and p an arbitrary, but fixed prime. We call a subset [x~]~ea of G, not containing 0, p-independent, if for any finite subset xl .... ,x~ a relation nlxl-[-... q-nkx1~ EprG
.IOSR Journal of Mathematics (IOSR-JM) , 2023
There is an existing theorem showing that not every group has a minimal generating set, by relying on a claim that all Proper Subgroups of infinite − Group (a group in which the order, () of its element is the power of some primes) are finite and cyclic. This paper shows as one of its objectives that not all Proper Subgroups of an infinitely generated − Group are finite and cyclic. Furthermore, the concepts of Maximal Independent sets and Minimal Generating Set are investigated for condition under which both concepts coincide. It is also shown that in the additive semigroup of integers, there are infinite minimal generating sets with different number of elements. This gives the implication that the dimension of vector spaces do not have analog in semigroups. Equivalently, these same examples serve as examples of infinite inequivalent maximal independent sets.
European Journal of Pure and Applied Mathematics
A nonempty set G is a g-group [with respect to a binary operation ∗] if it satisfies the following properties: (g1) a ∗ (b ∗ c) = (a ∗ b) ∗ c for all a, b, c ∈ G; (g2) for each a ∈ G, there exists an element e ∈ G such that a ∗ e = a = e ∗ a (e is called an identity element of a); and, (g3) for each a ∈ G, there exists an element b ∈ G such that a ∗ b = e = b ∗ a for some identity element eof a. In this study, we gave some important properties of g-subgroups, homomorphism of g-groups, andthe zero element. We also presented a couple of ways to construct g-groups and g-subgroups.
Study of Groups with basic property
Maǧallaẗ ǧāmiʻaẗ al-anbār li-l-ʻulūm al-ṣirfaẗ, 2012
The purpose of this paper is to study the concept of dependence , independence and the basis of some algebraic structure and give the definition of a finite group with basic property and study some of its basic properties .
A Simplicity Criterion for Finite Groups
Journal of Algebra, 1997
In this paper, we investigate the relation between the number of primes and that of Ž. composite numbers in G and obtain a criterion for the simplicity of finite e groups.
A sufficient conditon for solvability of finite groups
arXiv (Cornell University), 2017
The following theorem is proved: Let G be a finite group and π e (G) be the set of element orders in G. If π e (G) ∩ {2} = ∅; or π e (G) ∩ {3, 4} = ∅; or π e (G) ∩ {3, 5} = ∅, then G is solvable. Moreover, using the intersection with π e (G) being empty set to judge G is solvable or not, only the above three cases. 1 Introduction Let G be a finite group. We have two basic sets: |G| and π e (G). There are many famous works about |G| in the history of group theory. The set π e (G)