Difference between revisions of "Multiplier group"
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| − | + | ''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==== | ====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