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A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090890/s0908901.png" /> of subgroups (cf. [[Subgroup|Subgroup]]) of a [[Group|group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090890/s0908902.png" /> satisfying the following conditions: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090890/s0908903.png" /> contains the unit subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090890/s0908904.png" /> and the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090890/s0908905.png" /> itself; and 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090890/s0908906.png" /> is totally ordered by inclusion, i.e. for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090890/s0908907.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090890/s0908908.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090890/s0908909.png" /> either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090890/s09089010.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090890/s09089011.png" />. One says that two subgroups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090890/s09089012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090890/s09089013.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090890/s09089014.png" /> constitute a jump if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090890/s09089015.png" /> follows directly from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090890/s09089016.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090890/s09089017.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090890/s09089018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090890/s09089019.png" /> in this system, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090890/s09089020.png" /> is a [[Normal subgroup|normal subgroup]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090890/s09089021.png" />. The quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090890/s09089022.png" /> is called a factor of the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090890/s09089023.png" />. A subgroup system in which all members are normal subgroups of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090890/s09089024.png" /> 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. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090890/s09089025.png" /> is contained in the centre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090890/s09089026.png" /> for any jump <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090890/s09089027.png" />. A subnormal subgroup system is called solvable if all its factors are Abelian.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090890/s09089028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090890/s09089029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090890/s09089030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090890/s09089031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090890/s09089032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090890/s09089033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090890/s09089034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090890/s09089035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090890/s09089036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090890/s09089037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090890/s09089038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090890/s09089039.png" />, the Kurosh–Chernikov classes of:
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<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090890/s09089040.png" />-groups: There is a solvable subnormal subgroup system;
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A set  $  \mathfrak A $
 +
of subgroups (cf. [[Subgroup|Subgroup]]) of a [[Group|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 $
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either  $  A \subseteq B $
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or  $  B \subseteq A $.
 +
One says that two subgroups  $  A $
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and  $  A  ^  \prime  $
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from  $  \mathfrak A $
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constitute a jump if  $  A  ^  \prime  $
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follows directly from  $  A $
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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 $
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and  $  A  ^  \prime  $
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in this system,  $  A $
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is a [[Normal subgroup|normal subgroup]] in  $  A  ^  \prime  $.  
 +
The quotient group  $  A  ^  \prime  /A $
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is called a factor of the system  $  \mathfrak A $.
 +
A subgroup system in which all members are normal subgroups of a group  $  G $
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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.
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090890/s09089041.png" />-groups: There is a well-ordered ascending solvable subnormal subgroup system;
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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}  $,
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$  N $,
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the Kurosh–Chernikov classes of:
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090890/s09089042.png" />-groups: Any subnormal subgroup system in such a group can be refined to a solvable subnormal one;
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$  RN $-
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groups: There is a solvable subnormal subgroup system;
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090890/s09089043.png" />-groups: There is a solvable normal subgroup system;
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$  \overline{RN}\; {}  ^ {*} $-
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groups: There is a well-ordered ascending solvable subnormal subgroup system;
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090890/s09089044.png" />-groups: There is a well-ordered ascending solvable normal subgroup system;
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$  \overline{RN}\; $-
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groups: Any subnormal subgroup system in such a group can be refined to a solvable subnormal one;
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090890/s09089045.png" />-groups: Any normal subgroup system in such a group can be refined to a solvable normal one;
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$  RI $-
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groups: There is a solvable normal subgroup system;
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090890/s09089046.png" />-groups: There is a central subgroup system;
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$  RI  ^ {*} $-
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groups: There is a well-ordered ascending solvable normal subgroup system;
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090890/s09089047.png" />-groups: There is a well-ordered ascending central subgroup system;
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$  \overline{RI}\; $-
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groups: Any normal subgroup system in such a group can be refined to a solvable normal one;
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090890/s09089048.png" />-groups: There is a well-ordered descending central subgroup system;
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$  Z $-
 +
groups: There is a central subgroup system;
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090890/s09089049.png" />-groups: Any normal subgroup system of this group can be refined to a central one;
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$  ZA $-
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groups: There is a well-ordered ascending central subgroup system;
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090890/s09089050.png" />-groups: Through any subgroup of this group there passes a subgroup system;
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$  ZD $-
 +
groups: There is a well-ordered descending central subgroup system;
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090890/s09089051.png" />-groups: Through any subgroup of this group there passes a well-ordered ascending subnormal subgroup system.
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$  \overline{Z}\; $-
 +
groups: Any normal subgroup system of this group can be refined to a central one;
 +
 
 +
$  \widetilde{N}  $-
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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|subgroup series]].
 
A particular case of a subgroup system is a [[Subgroup series|subgroup series]].
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.G. Kurosh,  "The theory of groups" , '''1–2''' , Chelsea  (1955–1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.N. Chernikov,  "Groups with given properties of subgroup systems" , Moscow  (1980)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.I. Kargapolov,  J.I. [Yu.I. Merzlyakov] Merzljakov,  "Fundamentals of the theory of groups" , Springer  (1979)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.G. Kurosh,  "The theory of groups" , '''1–2''' , Chelsea  (1955–1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.N. Chernikov,  "Groups with given properties of subgroup systems" , Moscow  (1980)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.I. Kargapolov,  J.I. [Yu.I. Merzlyakov] Merzljakov,  "Fundamentals of the theory of groups" , Springer  (1979)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D.J.S. Robinson,  "Finiteness condition and generalized soluble groups" , '''1–2''' , Springer  (1972)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D.J.S. Robinson,  "Finiteness condition and generalized soluble groups" , '''1–2''' , Springer  (1972)</TD></TR></table>

Latest revision as of 08:24, 6 June 2020


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)

Comments

References

[a1] D.J.S. Robinson, "Finiteness condition and generalized soluble groups" , 1–2 , Springer (1972)
How to Cite This Entry:
Subgroup system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Subgroup_system&oldid=48892
This article was adapted from an original article by N.S. Romanovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article