Schur multiplier of abelian group is its exterior square
From Groupprops
Statement
Suppose is an abelian group. The Schur multiplier of
, denoted
, which is the same as the second homology group for trivial group action
is isomorphic to the group
, defined as the exterior square of
viewed as a
-module.
Related facts
Generalizations
Generalization is to ... | Statement | How this is a special case |
---|---|---|
arbitrary group instead of abelian group | Schur multiplier is kernel of commutator map homomorphism from exterior square to derived subgroup | when the group is an abelian group, the commutator map is the trivial map and hence the kernel of the homomorphism is the whole exterior square. |
nilpotent multiplier instead of Schur multiplier | nilpotent multiplier of abelian group is graded component of free Lie ring | The Schur multiplier is the ![]() ![]() ![]() |