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.

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 SL_n (n ≥ 3) and Sp_2n (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, vol. 147, Berlin, New York: Springer-Verlag, p. 77, ISBN 978-0-387-94248-3, MR 1282290, Zbl 0801.19001. Errata