View source for Multiplier group

<!--
m0655201.png
$#A+1 = 18 n = 0
$#C+1 = 18 : ~/encyclopedia/old_files/data/M065/M.0605520 Multiplier group,
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

''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|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|commutator subgroup]] of  $  G $).

====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>