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 $RN$, $\overline{RN}{}^{\ast }$, $\overline{RN}$, $RI$, $R{I}^{\ast }$, $\overline{RI}$, $Z$, $ZA$, $ZD$, $\bar{Z}$, $\tilde{N}$, $N$, the Kurosh–Chernikov classes of:

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

$\overline{RN}{}^{\ast }$- 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;

$R{I}^{\ast }$- 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;

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

$\tilde{N}$- 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.

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