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wreath product
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Definition/Summary
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| The wreath product of permutation group P and group A is a group whose elements are elements of P along with vectors of elements of A. If the permutation group is of permutations over n symbols, then the vectors of elements of A have length n. |
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Equations
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The wreath-product elements are (p,{a}), and the group operation may be defined as
[itex](p_1,\{a_1\}) \cdot (p_2,\{a_2\}) = (p_1 p_2, \{a_1\} \cdot (p_1 \{a_2\}))[/itex]
where [itex]p \{a\}[/itex] is the group action of p on {a} or the permutation of {a} by p.
The identity element is
(identity permutation, vector of identity element of A),
and the inverse of (p,{a}) is
[itex](p,{a})^{-1} = (p^{-1},p^{-1}\{a^{-1}\})[/itex]
The commutator subgroup's elements have the form (identity permutation,{a}), though that subgroup need not have all the elements in the group with that form. |
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Recent forum threads on wreath product
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Breakdown
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Mathematics
> Algebra
>> Group Theory
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Commentary
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