Mennicke symbol
# 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 Mennicke (1965). They were named by Bass, Milnor & Serre (1967), 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 $W_q$ 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 $W_q$ with values in a group C is a function (a, b) → $[^b$ $_a]$ from $W_q$ 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 $C_q$ such that any Mennicke symbol with values in C can be obtained by composing the universal Mennicke symbol with a unique homomorphism from $C_q$ to C.

