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wreath product

 Definition/Summary 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.

 Equations The wreath-product elements are (p,{a}), and the group operation may be defined as $(p_1,\{a_1\}) \cdot (p_2,\{a_2\}) = (p_1 p_2, \{a_1\} \cdot (p_1 \{a_2\}))$ where $p \{a\}$ 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 $(p,{a})^{-1} = (p^{-1},p^{-1}\{a^{-1}\})$ 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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 Breakdown Mathematics > Algebra >> Group Theory