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Calling Sequence
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WreathProduct( G, H, ... )
RegularWreathProduct( G, H, ... )
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Parameters
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G,H, ...
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two or more permutation groups
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Description
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Let and be permutation groups. The wreath product G H_ of by is a permutation group constructed as a semi-direct product of copies of (called the base group), where is the degree of , and the action of on the base group is the action of by permuting the copies of . Thus, the order of G H_ is equal to , and the degree of the wreath product is the product of the degrees of and .
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The regular wreath product of and is the wreath product in which is considered as a regular permutation group on itself. (This is also called the "standard wreath product".)
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The WreathProduct( G, H ) command returns a permutation group that is the wreath product G H_.
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If more than two groups are provided as input, then an iterated wreath product is constructed using the left associative rule. For example, WreathProduct( A, B, C, D ) returns ((A B) C) D_.
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The RegularWreathProduct( G, H ) command returns the regular wreath product of and . In this case, it is not required that be a permutation group, as a regular permutation representation of the finite group is used instead. (Here, may be either a Cayley table group or a finitely presented finite group, as well as a permutation group, which need not be itself regular.)
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Examples
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| (1) |
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Iterated wreath products appear naturally as Sylow subgroups of symmetric groups of prime power degree.
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Note that iterated wreath products grow quite rapidly.
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| (8) |
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| (9) |
The wreath product construction is not commutative; notice that even the order is different.
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Note that the regular wreath product is also different in this case, since the second argument here is not the regular permutation representation of the symmetric group.
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Since both and are transitive, so too is their wreath product.
However, in general, the wreath product is not primitive.
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Here we construct a wreath product with an intransitive second argument.
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The resulting group is not transitive.
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| (17) |
Here we construct a wreath product with an intransitive first argument.
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Again, the result is an intransitive group.
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| (19) |
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| (20) |
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| (26) |
Notice that, if a group is already regular, then its regular wreath product is isomorphic to the ordinary wreath product.
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| (27) |
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