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''of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090960/s0909601.png" />''
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{{MSC|20D30|20D35}}
  
A [[Subgroup series|subgroup series]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090960/s0909602.png" />,
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A subnormal series (or ''subinvariant series'') of a group $G$ is a [[subgroup series]]
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$$
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E = G_0 \le G_1 \le \cdots \le G_n = G
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$$
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in which each subgroup $G_i$ is a normal subgroup of $G_{i+1}$. The quotient groups $G_{i+1}/G_i$ are called ''factors'', and the number $n$ is called the length of the subnormal series. Infinite subnormal series have also been studied (see [[Subgroup system]]).  
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090960/s0909603.png" /></td> </tr></table>
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A subnormal series that cannot be refined further is called a ''[[composition series]]'', and its factors are called ''composition factors''. Any two subnormal series of a group have isomorphic refinements and in particular, any two composition series are isomorphic (see [[Jordan–Hölder theorem]]).
  
where each subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090960/s0909604.png" /> is a normal subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090960/s0909605.png" />. The quotient groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090960/s0909606.png" /> are called factors, and the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090960/s0909607.png" /> is called the length of the subnormal series. Infinite subnormal series have also been studied (see [[Subgroup system|Subgroup system]]). A subnormal series that cannot be refined further is called a composition series, and its factors are called composition factors.
 
  
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A '''subnormal subgroup''' (also ''subinvariant'', ''attainable'' or ''accessible'') of $G$ is a subgroup that appears in some subnormal series of $G$. To indicate the subnormality of a subgroup $H$ in a group $G$, the notation $H \lhd\!\lhd G$ is used.
  
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The property of a subgroup to be subnormal is transitive. An intersection of subnormal subgroups is again a subnormal subgroup. The subgroup generated by two subnormal subgroups need not be subnormal. A group $G$ all subgroups of which are subnormal satisfies the normalizer condition, i.e. all subgroups differ from their normalizers (cf. [[Normalizer of a subset]]). Such a group is therefore [[Locally nilpotent group|locally nilpotent]].
  
====Comments====
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A subnormal subgroup of $G$ that coincides with its commutator subgroup and whose quotient by its centre is simple is called a ''component'' of $G$. The product of all components of $G$ is known as the ''layer'' of $G$. It is an important [[characteristic subgroup]] of $G$ in the theory of finite simple groups, see e.g. [[#References|[6]]].
A subnormal series is also called a subinvariant series.
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Hall jr.,  "The theory of groups" , Macmillan  (1959)  pp. Sect. 8.4</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</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>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  A.G. Kurosh,  "The theory of groups" , '''1–2''' , Chelsea  (1955–1956)  (Translated from Russian)
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<TR><TD valign="top">[3]</TD> <TD valign="top">  M. Hall jr.,  "The theory of groups" , Macmillan  (1959)  pp. Sect. 8.4</TD></TR>
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<TR><TD valign="top">[4]</TD> <TD valign="top">  J.C. Lennox,  S.E. Stonehewer,  "Subnormal subgroups of groups" , Clarendon Press  (1987)</TD></TR>
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<TR><TD valign="top">[5]</TD> <TD valign="top">  D.J.S. Robinson,  "A course in the theory of groups" , Springer  (1982)</TD></TR>
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<TR><TD valign="top">[6]</TD> <TD valign="top">  M. Suzuki,  "Group theory" , '''1–2''' , Springer  (1986)</TD></TR>
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</table>
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{{TEX|done}}

Latest revision as of 10:03, 3 January 2021

2010 Mathematics Subject Classification: Primary: 20D30 Secondary: 20D35 [MSN][ZBL]

A subnormal series (or subinvariant series) of a group $G$ is a subgroup series $$ E = G_0 \le G_1 \le \cdots \le G_n = G $$ in which each subgroup $G_i$ is a normal subgroup of $G_{i+1}$. The quotient groups $G_{i+1}/G_i$ are called factors, and the number $n$ is called the length of the subnormal series. Infinite subnormal series have also been studied (see Subgroup system).

A subnormal series that cannot be refined further is called a composition series, and its factors are called composition factors. Any two subnormal series of a group have isomorphic refinements and in particular, any two composition series are isomorphic (see Jordan–Hölder theorem).


A subnormal subgroup (also subinvariant, attainable or accessible) of $G$ is a subgroup that appears in some subnormal series of $G$. To indicate the subnormality of a subgroup $H$ in a group $G$, the notation $H \lhd\!\lhd G$ is used.

The property of a subgroup to be subnormal is transitive. An intersection of subnormal subgroups is again a subnormal subgroup. The subgroup generated by two subnormal subgroups need not be subnormal. A group $G$ all subgroups of which are subnormal satisfies the normalizer condition, i.e. all subgroups differ from their normalizers (cf. Normalizer of a subset). Such a group is therefore locally nilpotent.

A subnormal subgroup of $G$ that coincides with its commutator subgroup and whose quotient by its centre is simple is called a component of $G$. The product of all components of $G$ is known as the layer of $G$. It is an important characteristic subgroup of $G$ in the theory of finite simple groups, see e.g. [6].

References

[1] M.I. Kargapolov, J.I. [Yu.I. Merzlyakov] Merzljakov, "Fundamentals of the theory of groups" , Springer (1979) (Translated from Russian)
[2] A.G. Kurosh, "The theory of groups" , 1–2 , Chelsea (1955–1956) (Translated from Russian)
[3] M. Hall jr., "The theory of groups" , Macmillan (1959) pp. Sect. 8.4
[4] J.C. Lennox, S.E. Stonehewer, "Subnormal subgroups of groups" , Clarendon Press (1987)
[5] D.J.S. Robinson, "A course in the theory of groups" , Springer (1982)
[6] M. Suzuki, "Group theory" , 1–2 , Springer (1986)
How to Cite This Entry:
Subnormal series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Subnormal_series&oldid=19288
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article