Multiplier group

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)