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)