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Subgroup system

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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)
How to Cite This Entry:
Subgroup system. N.S. Romanovskii (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Subgroup_system&oldid=11545
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098