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A counterexample to Zarrin’s conjecture on sizes of finite nonabelian simple groups in relation to involution sizes
- Source :
- Archiv der Mathematik. 112:225-226
- Publication Year :
- 2018
- Publisher :
- Springer Science and Business Media LLC, 2018.
-
Abstract
- Let $$I_n(G)$$ denote the number of elements of order n in a finite group G. In 1979, Herzog (Proc Am Math Soc 77:313–314, 1979) conjectured that two finite simple groups containing the same number of involutions have the same order. In a 2018 paper (Arch Math 111:349–351, 2018), Zarrin disproved Herzog’s conjecture with a counterexample. Then he conjectured that “if S is a non-abelian simple group and G a group such that $$I_2(G)=I_2(S)$$ and $$I_p(G) =I_p(S)$$ for some odd prime divisor p, then $$|G|=|S|$$ ”. In this paper, we give more counterexamples to Herzog’s conjecture. Moreover, we disprove Zarrin’s conjecture.
- Subjects :
- Involution (mathematics)
Finite group
Conjecture
Mathematics::Commutative Algebra
General Mathematics
010102 general mathematics
01 natural sciences
Combinatorics
Simple group
0103 physical sciences
Prime factor
010307 mathematical physics
Classification of finite simple groups
0101 mathematics
Mathematics
Counterexample
Subjects
Details
- ISSN :
- 14208938 and 0003889X
- Volume :
- 112
- Database :
- OpenAIRE
- Journal :
- Archiv der Mathematik
- Accession number :
- edsair.doi...........7af8b44e515b526c4c6f3668ec17ca59