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''multiplicator, of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065520/m0655201.png" /> represented as a quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065520/m0655202.png" /> of a free group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065520/m0655203.png" />''
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The quotient group
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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/m/m065/m065520/m0655204.png" /></td> </tr></table>
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''multiplicator, of a group  $  G $
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represented as a quotient group  $  F / R $
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of a free group  $  F $''
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065520/m0655205.png" /> is the [[Commutator subgroup|commutator subgroup]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065520/m0655206.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065520/m0655207.png" /> is the mutual commutator subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065520/m0655208.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065520/m0655209.png" />. The multiplicator of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065520/m06552010.png" /> does not depend on the way in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065520/m06552011.png" /> is presented as a quotient group of a free group. It is isomorphic to the second homology group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065520/m06552012.png" /> with integer coefficients. In certain branches of group theory the question of non-triviality of the multiplicator of a group is important.
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The quotient group
  
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$$
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R \cap F ^ { \prime } / [ R , F ] ,
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$$
  
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where  $  F ^ { \prime } $
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is the [[Commutator subgroup|commutator subgroup]] of  $  F $
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and  $  [ R , F ] $
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is the mutual commutator subgroup of  $  R $
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and  $  F $.
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The multiplicator of  $  G $
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does not depend on the way in which  $  G $
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is presented as a quotient group of a free group. It is isomorphic to the second homology group of  $  G $
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with integer coefficients. In certain branches of group theory the question of non-triviality of the multiplicator of a group is important.
  
 
====Comments====
 
====Comments====
The usual name in the Western literature is Schur multiplier (or multiplicator). It specifically enters in the study of central extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065520/m06552013.png" /> and in the study of perfect groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065520/m06552014.png" /> (i.e. groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065520/m06552015.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065520/m06552016.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065520/m06552017.png" /> is the [[Commutator subgroup|commutator subgroup]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065520/m06552018.png" />).
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The usual name in the Western literature is Schur multiplier (or multiplicator). It specifically enters in the study of central extensions of $  G $
 +
and in the study of perfect groups $  G $(
 +
i.e. groups $  G $
 +
for which $  G = [ G, G] $,  
 +
where $  [ G, G] $
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is the [[Commutator subgroup|commutator subgroup]] of $  G $).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D.J.S. Robinson,  "A course in the theory of groups" , Springer  (1980)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D.J.S. Robinson,  "A course in the theory of groups" , Springer  (1980)</TD></TR></table>

Latest revision as of 08:02, 6 June 2020


multiplicator, of a group $ G $ represented as a quotient group $ F / R $ of a free group $ F $

The quotient group

$$ R \cap F ^ { \prime } / [ R , F ] , $$

where $ F ^ { \prime } $ is the commutator subgroup of $ F $ and $ [ R , F ] $ is the mutual commutator subgroup of $ R $ and $ F $. The multiplicator of $ G $ does not depend on the way in which $ G $ is presented as a quotient group of a free group. It is isomorphic to the second homology group of $ G $ with integer coefficients. In certain branches of group theory the question of non-triviality of the multiplicator of a group is important.

Comments

The usual name in the Western literature is Schur multiplier (or multiplicator). It specifically enters in the study of central extensions of $ G $ and in the study of perfect groups $ G $( i.e. groups $ G $ for which $ G = [ G, G] $, where $ [ G, G] $ is the commutator subgroup of $ G $).

References

[a1] D.J.S. Robinson, "A course in the theory of groups" , Springer (1980)
How to Cite This Entry:
Multiplier group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Multiplier_group&oldid=13653
This article was adapted from an original article by A.L. Shmel'kin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article