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), 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) → [ba] from Wq to C such that
- [01] = 1, [bca] = [ba][ca]
- [ba] = [b + taa] if t is in q, [ba] = [ba + 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.
References
- Bass, Hyman (1968), Algebraic K-theory, Mathematics Lecture Note Series, New York-Amsterdam: W.A. Benjamin, Inc., pp. 279–342
- Bass, Hyman; Milnor, John Willard; Serre, Jean-Pierre (1967), "Solution of the congruence subgroup problem for SLn (n ≥ 3) and Sp2n (n ≥ 2)", Publications Mathématiques de l'IHÉS (33): 59–137, doi:10.1007/BF02684586, ISSN 1618-1913, http://www.numdam.org/item?id=PMIHES_1967__33__59_0 Erratum
- Mennicke, Jens L. (1965), "Finite factor groups of the unimodular group", Annals of Mathematics, Second Series 81 (1): 31–37, doi:10.2307/1970380, ISSN 0003-486X
- Rosenberg, Jonathan (1994), Algebraic K-theory and its applications, Graduate Texts in Mathematics, 147, Berlin, New York: Springer-Verlag, p. 77, ISBN 978-0-387-94248-3. Errata
 |  |
