Subgroup system

A set $ \mathfrak A $ of subgroups (cf. Subgroup) of a group $ G $ satisfying the following conditions: 1) $ \mathfrak A $ contains the unit subgroup $ 1 $ and the group $ G $ itself; and 2) $ \mathfrak A $ is totally ordered by inclusion, i.e. for any $ A $ and $ B $ from $ \mathfrak A $ either $ A \subseteq B $ or $ B \subseteq A $. One says that two subgroups $ A $ and $ A ^ \prime $ from $ \mathfrak A $ constitute a jump if $ A ^ \prime $ follows directly from $ A $ in $ \mathfrak A $. 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 $ A $ and $ A ^ \prime $ in this system, $ A $ is a normal subgroup in $ A ^ \prime $. The quotient group $ A ^ \prime /A $ is called a factor of the system $ \mathfrak A $. A subgroup system in which all members are normal subgroups of a group $ G $ 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. $ A ^ \prime /A $ is contained in the centre of $ G/A $ for any jump $ A, A ^ \prime $. 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 $ RN $, $ \overline{RN}\; {} ^ {*} $, $ \overline{RN}\; $, $ RI $, $ RI ^ {*} $, $ \overline{RI}\; $, $ Z $, $ ZA $, $ ZD $, $ \overline{Z}\; $, $ \widetilde{N} $, $ N $, the Kurosh–Chernikov classes of:

$ RN $- groups: There is a solvable subnormal subgroup system;

$ \overline{RN}\; {} ^ {*} $- groups: There is a well-ordered ascending solvable subnormal subgroup system;

$ \overline{RN}\; $- groups: Any subnormal subgroup system in such a group can be refined to a solvable subnormal one;

$ RI $- groups: There is a solvable normal subgroup system;

$ RI ^ {*} $- groups: There is a well-ordered ascending solvable normal subgroup system;

$ \overline{RI}\; $- groups: Any normal subgroup system in such a group can be refined to a solvable normal one;

$ Z $- groups: There is a central subgroup system;

$ ZA $- groups: There is a well-ordered ascending central subgroup system;

$ ZD $- groups: There is a well-ordered descending central subgroup system;

$ \overline{Z}\; $- groups: Any normal subgroup system of this group can be refined to a central one;

$ \widetilde{N} $- groups: Through any subgroup of this group there passes a subgroup system;

$ N $- 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)

References

[a1]	D.J.S. Robinson, "Finiteness condition and generalized soluble groups" , 1–2 , Springer (1972)