Difference between revisions of "Multiplier group"
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| − | + | ''multiplicator, of a group | |
| + | 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==== | ====Comments==== | ||
| − | The usual name in the Western literature is Schur multiplier (or multiplicator). It specifically enters in the study of central extensions of | + | 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==== | ====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=47939
Multiplier group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Multiplier_group&oldid=47939
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