Subgroup system
A set of subgroups (cf. Subgroup) of a group
satisfying the following conditions: 1)
contains the unit subgroup
and the group
itself; and 2)
is totally ordered by inclusion, i.e. for any
and
from
either
or
. One says that two subgroups
and
from
constitute a jump if
follows directly from
in
. A subgroup system that is closed with respect to union and intersection is called complete. A complete subgroup system is called subnormal if for any jump
and
in this system,
is a normal subgroup in
. The quotient group
is called a factor of the system
. A subgroup system in which all members are normal subgroups of a group
is called normal. In the case where one subnormal system contains another (in the set-theoretical sense), the first is called a refinement of the second. A normal subgroup system is called central if all its factors are central, i.e.
is contained in the centre of
for any jump
. A subnormal subgroup system is called solvable if all its factors are Abelian.
The presence in a group of some subgroup system enables one to distinguish various subclasses in the class of all groups, of which the ones most used are ,
,
,
,
,
,
,
,
,
,
,
, the Kurosh–Chernikov classes of:
-groups: There is a solvable subnormal subgroup system;
-groups: There is a well-ordered ascending solvable subnormal subgroup system;
-groups: Any subnormal subgroup system in such a group can be refined to a solvable subnormal one;
-groups: There is a solvable normal subgroup system;
-groups: There is a well-ordered ascending solvable normal subgroup system;
-groups: Any normal subgroup system in such a group can be refined to a solvable normal one;
-groups: There is a central subgroup system;
-groups: There is a well-ordered ascending central subgroup system;
-groups: There is a well-ordered descending central subgroup system;
-groups: Any normal subgroup system of this group can be refined to a central one;
-groups: Through any subgroup of this group there passes a subgroup system;
-groups: Through any subgroup of this group there passes a well-ordered ascending subnormal subgroup system.
A particular case of a subgroup system is a subgroup series.
References
[1] | A.G. Kurosh, "The theory of groups" , 1–2 , Chelsea (1955–1956) (Translated from Russian) |
[2] | S.N. Chernikov, "Groups with given properties of subgroup systems" , Moscow (1980) (In Russian) |
[3] | M.I. Kargapolov, J.I. [Yu.I. Merzlyakov] Merzljakov, "Fundamentals of the theory of groups" , Springer (1979) (Translated from Russian) |
Comments
References
[a1] | D.J.S. Robinson, "Finiteness condition and generalized soluble groups" , 1–2 , Springer (1972) |
Subgroup system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Subgroup_system&oldid=48892