TY - JOUR

T1 - On finite groups generated by strongly cosubnormal subgroups

AU - Ballester-Bolinches, A.

AU - Cossey, John

AU - Esteban-Romero, R.

PY - 2003/1/1

Y1 - 2003/1/1

N2 - Two subgroups A and B of a group G are cosubnormal if A and B are subnormal in their join (A, B) and are strongly cosubnormal if every subgroup of A is cosubnormal with every subgroup of B. We find necessary and sufficient conditions for A and B to be strongly cosubnormal in (A, B) and, if Z is the hypercentre of G = (A, B), we show that A and B are strongly cosubnormal if and only if G/Z is the direct product of AZ/Z and BZ/Z. We also show that projectors and residuals for certain formations can easily be constructed in such a group. Two subgroups A and B of a group G are N-connected if every cyclic subgroup of A is cosubnormal with every cyclic subgroup of B (N denotes the class of nilpotent groups). Though the concepts of strong cosubnormality and N-connectedness are clearly closely related, we give an example to show that they are not equivalent. We note, however, that if G is the product of the N-connected subgroups A and B, then A and B are strongly cosubnormal.

AB - Two subgroups A and B of a group G are cosubnormal if A and B are subnormal in their join (A, B) and are strongly cosubnormal if every subgroup of A is cosubnormal with every subgroup of B. We find necessary and sufficient conditions for A and B to be strongly cosubnormal in (A, B) and, if Z is the hypercentre of G = (A, B), we show that A and B are strongly cosubnormal if and only if G/Z is the direct product of AZ/Z and BZ/Z. We also show that projectors and residuals for certain formations can easily be constructed in such a group. Two subgroups A and B of a group G are N-connected if every cyclic subgroup of A is cosubnormal with every cyclic subgroup of B (N denotes the class of nilpotent groups). Though the concepts of strong cosubnormality and N-connectedness are clearly closely related, we give an example to show that they are not equivalent. We note, however, that if G is the product of the N-connected subgroups A and B, then A and B are strongly cosubnormal.

UR - http://www.scopus.com/inward/record.url?scp=0037283136&partnerID=8YFLogxK

U2 - 10.1016/S0021-8693(02)00535-5

DO - 10.1016/S0021-8693(02)00535-5

M3 - Article

SN - 0021-8693

VL - 259

SP - 226

EP - 234

JO - Journal of Algebra

JF - Journal of Algebra

IS - 1

ER -