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 (a, b) 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 (a, b) → [$b a$] from Wq to C such that
 * [$0 1$] = 1, [$bc a$] = [$b a$][$c a$]
 * [$b a$] = [$b + ta a$] if t is in q, [$b a$] = [$b a + tb$] 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.