Wreath product
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Wreath product | |
---|---|
Symbol | [1] |
Size formula | [2] |
Algebraic properties | |
Algebraic structure | Unital magma |
Associative | No |
Commutative | No |
Identity | S0 |
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]
- ↑ 1.0 1.1 1.2 McMahan, Peter (2003). Wreath-Product Polytopes (PDF) (Thesis). Reed College.
- ↑ Here n is determined by .