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*) → [^{\( \frac {b}{a} \)}] from *W*_{q} 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 *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*.

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

Undergraduate Texts in Mathematics

Graduate Studies in Mathematics

Hellenica World - Scientific Library

Retrieved from "http://en.wikipedia.org/"

All text is available under the terms of the GNU Free Documentation License