Information AboutMultiplicator |
| CATEGORIES ABOUT SCHUR MULTIPLIER | |
| group theory | |
| homological algebra | |
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:. If the group is presented in terms of a Free Group ''F'' on a set of generators, and a Normal Subgroup ''R'' generated by a set of relations on the generators, so that :, then by Hopf's integral homology formula we have that the Schur multiplier is :, where B is the group generated by the Commutator s aba for a in A and b in B. It can also be expressed in terms of cohomology, as : where ''G'' acts trivially on the multiplicative group of the nonzero complex numbers. Schur multipliers are of especial interest when ''G'' is a Perfect Group (a group equal to its own commutator subgroup). A perfect group has a unique perfect maximal Central Extension , or universal covering group, whose center contains the Schur multiplier and whose quotient by it is the perfect group. The Schur multiplier is due to Issai Schur , and can be said to represent the beginning of Group Cohomology (or at least of the higher groups ''H''2 etc.). REFERENCES
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