Schur Multiplier: Multiplier, Mathematics, Group Theory, Finite Group, Issai Schur, Cohomology, Abelian Sylow Theorems, Dihedral of Order 6

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, more specifically in group theory, the Schur multiplier is an important invariant of a finite group that has applications in many areas of mathematics. The Schur multiplier is sometimes called the multiplicator and is named after Issai Schur. It is often defined as the second homology group of a group G with coefficients in the integers. For instance, the Schur multiplier of the nonabelian group of order 6 is the trivial group since every Sylow subgroup is cyclic. The Schur multiplier of the elementary abelian group of order 16 is an elementary abelian group of order 64, showing that the multiplier can be strictly larger than the group itself. The Schur multiplier of the quaternion group is trivial, but the Schur multiplier of dihedral 2-groups has order 2.