TY - JOUR

T1 - On Prime Spectrum of Maximal Subgroups in Finite Groups

AU - Zhang, Chi

AU - Guo, Wenbin

AU - Maslova, Natalia V.

AU - Revin, Danila O.

PY - 2018/12/1

Y1 - 2018/12/1

N2 - For a positive integer n, we denote by ?n the set of all prime divisors of n. For a finite group G, the set ?G:= ?|G| is called the prime spectrum of G. Let M?G mean that M is a maximal subgroup of G. We put KG = max{|?G\?M |: M ? G} and kG = min{|?G\?M |: M ? G}. In this notice, using well-known number-theoretical results, we present a number of examples to show that both KG and kG are unbounded in general. This implies that the problem "Are kG and KG bounded by some constant k?", raised by Monakhov and Skiba in 2016, is solved in the negative.

AB - For a positive integer n, we denote by ?n the set of all prime divisors of n. For a finite group G, the set ?G:= ?|G| is called the prime spectrum of G. Let M?G mean that M is a maximal subgroup of G. We put KG = max{|?G\?M |: M ? G} and kG = min{|?G\?M |: M ? G}. In this notice, using well-known number-theoretical results, we present a number of examples to show that both KG and kG are unbounded in general. This implies that the problem "Are kG and KG bounded by some constant k?", raised by Monakhov and Skiba in 2016, is solved in the negative.

KW - finite group

KW - finite simple group

KW - maximal subgroup

KW - prime spectrum

KW - NONABELIAN COMPOSITION FACTORS

UR - http://www.scopus.com/inward/record.url?scp=85057727299&partnerID=8YFLogxK

U2 - 10.1142/S1005386718000408

DO - 10.1142/S1005386718000408

M3 - Article

AN - SCOPUS:85057727299

VL - 25

SP - 579

EP - 584

JO - Algebra Colloquium

JF - Algebra Colloquium

SN - 1005-3867

IS - 4

ER -