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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 . 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
This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.
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