Mennicke symbol


In mathematics, a Mennicke symbol is a map from pairs of elements of a number field to an abelian group satisfying some identities found by. They were named by, who used them in their solution of the congruence subgroup problem.

Definition

Suppose that A is a Dedekind domain and q is a non-zero ideal of A. The set Wq is defined to be the set of pairs with a = 1 mod q, b = 0 mod q, such that a and b generate the unit ideal.
A Mennicke symbol on Wq with values in a group C is a function → from Wq to C such that
  • = 1, =
  • = if t is in q, = if t is in A.
There is a universal Mennicke symbol with values in a group Cq such that any Mennicke symbol with values in C can be obtained by composing the universal Mennicke symbol with a unique homomorphism from Cq to C.