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)