# Wreath product

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Wreath product | |
---|---|

Symbol | ^{[1]} |

Size formula | ^{[2]} |

Algebraic properties | |

Algebraic structure | Unital magma |

Associative | No |

Commutative | No |

Identity | S_{0} |

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 S
_{mn}.^{[1]}

## References[edit | edit source]

- ↑
^{1.0}^{1.1}^{1.2}McMahan, Peter (2003).*Wreath-Product Polytopes*(PDF) (Thesis). Reed College. - ↑ Here n is determined by .