Wreath product

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Wreath product
Symbol[1]
Size formula[2]
Algebraic properties
Algebraic structureUnital magma
AssociativeNo
CommutativeNo
IdentityS0


The wreath product is an product that operates on groups.

Definition[edit | edit source]

Given a group and a permutation group such that , then is a group where:

and:

Put simply the operation permutes the tuple g in the first argument by the permutation h' in the second argument, and then combines the two tuples pairwise with the usual operations.[1]

This definition can be generalized to apply to arbitrary groups, by observing that due to Cayley's theorem every group is isomorphic to a permutation group.

Properties[edit | edit source]

  • For groups G and : .
  • If and then is isomorphic to a subgroup of Smn.[1]

References[edit | edit source]

  1. 1.0 1.1 1.2 McMahan, Peter (2003). Wreath-Product Polytopes (PDF) (Thesis). Reed College.
  2. Here n is determined by .