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Multiplier group

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multiplicator, of a group represented as a quotient group of a free group

The quotient group

where is the commutator subgroup of and is the mutual commutator subgroup of and . The multiplicator of does not depend on the way in which is presented as a quotient group of a free group. It is isomorphic to the second homology group of 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 and in the study of perfect groups (i.e. groups for which , where is the commutator subgroup of ).

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
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