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 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) → [\( \frac {b}{a} \)] from Wq to C such that
- [\( \frac {0}{1} \)] = 1, [\( \frac {bc}{a} \)] = [\( \frac {b}{a} \)][\( \frac {c}{a} \)]
- [\( \frac {b}{a} \)] = [\( \frac {b + ta}{a} \)] if t is in q, [\( \frac {b}{a} \)] = [\( \frac {b}{a+tb} \)] 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.
References
Bass, Hyman (1968), Algebraic K-theory, Mathematics Lecture Note Series, New York-Amsterdam: W.A. Benjamin, Inc., pp. 279–342, Zbl 0174.30302
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, MR 0244257 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, JSTOR 1970380, MR 0171856
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, MR 1282290, Zbl 0801.19001. Errata
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