TY - JOUR

T1 - On supersolvable groups and the nilpotator

AU - Csörgö, Piroska

AU - Herzog, Marcel

N1 - Funding Information:
The first author was partly supported by the Hungarian National Foundation for Scientific Research, Grants No. T 038059 and T 034878.

PY - 2004

Y1 - 2004

N2 - A finite group G is called G a J-group if each subnormal subgroup of G is normal in G and a subgroup K of G is called an ℋ -subgroup of G if N G (K) n Kg ∩ Kg⊆ K for all g εG. Using the notion of ℋ-subgroups, we present some new conditions for supersolvability and we characterize supersolvable groups, which are either J-groups or A-groups (i.e., all their Sylow subgroups are abelian). For example, we prove that if all cyclic subgroups of G of prime order or of order 4 are ℋsubgroups of G, then G is supersolvable with a well defined structure. We also show, that an A-group G is supersolvable if and only if its Sylow subgroups are products of cyclic ℋ-subgroups of G.

AB - A finite group G is called G a J-group if each subnormal subgroup of G is normal in G and a subgroup K of G is called an ℋ -subgroup of G if N G (K) n Kg ∩ Kg⊆ K for all g εG. Using the notion of ℋ-subgroups, we present some new conditions for supersolvability and we characterize supersolvable groups, which are either J-groups or A-groups (i.e., all their Sylow subgroups are abelian). For example, we prove that if all cyclic subgroups of G of prime order or of order 4 are ℋsubgroups of G, then G is supersolvable with a well defined structure. We also show, that an A-group G is supersolvable if and only if its Sylow subgroups are products of cyclic ℋ-subgroups of G.

KW - J-groups

KW - Supersolvable groups

KW - The nilpotator and the supersolvator of a group

KW - ℋ-groups

UR - http://www.scopus.com/inward/record.url?scp=11244296764&partnerID=8YFLogxK

U2 - 10.1081/AGB-120027916

DO - 10.1081/AGB-120027916

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AN - SCOPUS:11244296764

SN - 0092-7872

VL - 32

SP - 609

EP - 620

JO - Communications in Algebra

JF - Communications in Algebra

IS - 2

ER -