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 . They were named by , 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'') → [] from ''W''_''q'' to ''C'' such that *[] = 1, [] = [][] *[] = [] if ''t'' is in ''q'', [] = [] 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

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Erratum
* *{{Citation , last1=Rosenberg , first1=Jonathan , title=Algebraic K-theory and its applications , publisher= Springer-Verlag , location=Berlin, New York , series= Graduate Texts in Mathematics , isbn=978-0-387-94248-3 , mr=1282290 , zbl=0801.19001 , year=1994 , volume=147 , page=77.
Errata
Group theory Algebraic K-theory