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Difference between revisions of "Subnormal series"

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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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''of a group $G$''
  
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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 [[subgroup series]] of $G$,
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$$
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090960/s0909603.png" /></td> </tr></table>
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E = G_0 \le G_1 \le \cdots \le G_n = G
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$$
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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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where 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''.
  
  
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====References====
 
====References====
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<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">[a1]</TD> <TD valign="top">  M. Hall jr.,  "The theory of groups" , Macmillan  (1959)  pp. Sect. 8.4</TD></TR>
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Revision as of 11:06, 1 March 2018

of a group $G$

A subgroup series of $G$, $$ E = G_0 \le G_1 \le \cdots \le G_n = G $$ where 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.


Comments

A subnormal series is also called a subinvariant series.

References

[a1] M. Hall jr., "The theory of groups" , Macmillan (1959) pp. Sect. 8.4
How to Cite This Entry:
Subnormal series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Subnormal_series&oldid=42878
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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