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Some numerical results on groups

Some numerical results on groups Acta Mathematica Academiae Scientiarum Hungaricae Tomus 26 (1--2), (1975), 91--96. By H. S. FINKELSTEIN (Atlanta) In 1893 FROBENIUS [2] began an investigation of numerical properties associated with the solutions of the equation x"= 1 in a finite group. Then in 1925 WEISNER [8] obtained a divisor for the number of elements whose order is divisible by an integer k, if this number is not zero. The purpose of this paper is to consider a set of related group elements and investigate numerical relations which imply certain group-theoretic properties, namely cyclicity. I. Notation Throughout this paper G will denote a finite group of order IGI. If gEG, let o(g) denote the order of the element g. Let His, k]= {xEG: klo(x)[sk} where alb means a divides b, and let h[s, k]= IN[s, k]l, the size of the set H[s, k]. Let m~k= =max {d: disk, (d, k)=l}. If IGl=ft and (p, t)=l we write F/IGI. Let (p denote Euler's phi-function. [1. Divisibility conditions TURKIN [7] and DUBUQUE [1] considered the set H[s, k] and proved msk[h[s, k] and cp(k)lh[s, k] respectively. In general, the product rns~o(k ) is not a divisor of his, kl. LEMMA 1. G is cyclic if http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematica Academiae Scientiarum Hungarica Springer Journals

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References (6)

Publisher
Springer Journals
Copyright
Copyright
Subject
Mathematics; Mathematics, general
ISSN
0001-5954
eISSN
1588-2632
DOI
10.1007/BF01895952
Publisher site
See Article on Publisher Site

Abstract

Acta Mathematica Academiae Scientiarum Hungaricae Tomus 26 (1--2), (1975), 91--96. By H. S. FINKELSTEIN (Atlanta) In 1893 FROBENIUS [2] began an investigation of numerical properties associated with the solutions of the equation x"= 1 in a finite group. Then in 1925 WEISNER [8] obtained a divisor for the number of elements whose order is divisible by an integer k, if this number is not zero. The purpose of this paper is to consider a set of related group elements and investigate numerical relations which imply certain group-theoretic properties, namely cyclicity. I. Notation Throughout this paper G will denote a finite group of order IGI. If gEG, let o(g) denote the order of the element g. Let His, k]= {xEG: klo(x)[sk} where alb means a divides b, and let h[s, k]= IN[s, k]l, the size of the set H[s, k]. Let m~k= =max {d: disk, (d, k)=l}. If IGl=ft and (p, t)=l we write F/IGI. Let (p denote Euler's phi-function. [1. Divisibility conditions TURKIN [7] and DUBUQUE [1] considered the set H[s, k] and proved msk[h[s, k] and cp(k)lh[s, k] respectively. In general, the product rns~o(k ) is not a divisor of his, kl. LEMMA 1. G is cyclic if

Journal

Acta Mathematica Academiae Scientiarum HungaricaSpringer Journals

Published: May 21, 2016

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