scholarly journals The Closures of Wreath Products in Product Action

2021 ◽  
Author(s):  
A. V. Vasil’ev ◽  
I. N. Ponomarenko
Keyword(s):  
Wreath Products ◽  
Product Action

scholarly journals EMBEDDING PERMUTATION GROUPS INTO WREATH PRODUCTS IN PRODUCT ACTION

2012 ◽  
Vol 92 (1) ◽  
pp. 127-136 ◽  
Author(s):  
CHERYL E. PRAEGER ◽  
CSABA SCHNEIDER
Keyword(s):  
Permutation Group ◽  
Wreath Product ◽  
Automorphism Groups ◽  
Wreath Products ◽  
Error Correcting Codes ◽  
Permutation Groups ◽  
Graph Products ◽  
Product Action ◽  
Hamming Graphs

AbstractWe consider the wreath product of two permutation groups G≤Sym Γ and H≤Sym Δ as a permutation group acting on the set Π of functions from Δ to Γ. Such groups play an important role in the O’Nan–Scott theory of permutation groups and they also arise as automorphism groups of graph products and codes. Let X be a subgroup of Sym Γ≀Sym Δ. Our main result is that, in a suitable conjugate of X, the subgroup of SymΓ induced by a stabiliser of a coordinate δ∈Δ only depends on the orbit of δ under the induced action of X on Δ. Hence, if X is transitive on Δ, then X can be embedded into the wreath product of the permutation group induced by the stabiliser Xδ on Γ and the permutation group induced by X on Δ. We use this result to describe the case where X is intransitive on Δ and offer an application to error-correcting codes in Hamming graphs.

scholarly journals Wreath decompositions of finite permutation groups

1989 ◽  
Vol 40 (2) ◽  
pp. 255-279 ◽  
Author(s):  
L. G. Kovács
Keyword(s):  
Permutation Group ◽  
Wreath Product ◽  
Transitive Group ◽  
Wreath Products ◽  
Permutation Groups ◽  
Transitive Permutation Group ◽  
Formal Definition ◽  
The Third ◽  
Product Action ◽  
Factor Decomposition

There is a familiar construction with two finite, transitive permutation groups as input and a finite, transitive permutation group, called their wreath product, as output. The corresponding ‘imprimitive wreath decomposition’ concept is the first subject of this paper. A formal definition is adopted and an overview obtained for all such decompositions of any given finite, transitive group. The result may be heuristically expressed as follows, exploiting the associative nature of the construction. Each finite transitive permutation group may be written, essentially uniquely, as the wreath product of a sequence of wreath-indecomposable groups, amid the two-factor wreath decompositions of the group are precisely those which one obtains by bracketing this many-factor decomposition.If both input groups are nontrivial, the output above is always imprimitive. A similar construction gives a primitive output, called the wreath product in product action, provided the first input group is primitive and not regular. The second subject of the paper is the ‘product action wreath decomposition’ concept dual to this. An analogue of the result stated above is established for primitive groups with nonabelian socle.Given a primitive subgroup G with non-regular socle in some symmetric group S, how many subgroups W of S which contain G and have the same socle, are wreath products in product action? The third part of the paper outlines an algorithm which reduces this count to questions about permutation groups whose degrees are very much smaller than that of G.

scholarly journals Innately transitive subgroups of wreath products in product action

2005 ◽  
Vol 358 (4) ◽  
pp. 1619-1641 ◽  
Author(s):  
Robert W. Baddeley ◽  
Cheryl E. Praeger ◽  
Csaba Schneider
Keyword(s):  
Wreath Products ◽  
Product Action

scholarly journals Hereditarily just infinite profinite groups with complete Hausdorff dimension spectrum

2019 ◽  
Vol 18 (11) ◽  
pp. 1950216
Author(s):  
Yiftach Barnea ◽  
Matteo Vannacci
Keyword(s):  
Hausdorff Dimension ◽  
Wreath Products ◽  
Inverse Limits ◽  
Normal Subgroups ◽  
Profinite Groups ◽  
Dimension Spectrum ◽  
Product Action ◽  
Hausdorff Dimension Spectrum

We prove that the inverse limits of certain iterated wreath products in product action have complete Hausdorff dimension spectrum with respect to their unique maximal filtration of open normal subgroups. Moreover we can produce explicitly subgroups with a specified Hausdorff dimension.

scholarly journals Transitive simple subgroups of wreath products in product action

2004 ◽  
Vol 77 (1) ◽  
pp. 55-72 ◽  
Author(s):  
Robert W. Baddeley ◽  
Cheryl E. Praeger ◽  
Csaba Schneider
Keyword(s):  
Symmetric Group ◽  
Wreath Product ◽  
Detailed Examination ◽  
Wreath Products ◽  
Permutation Groups ◽  
Simple Permutation ◽  
Product Action ◽  
Simple Subgroup ◽  
Finite Symmetric Group

AbstractA transitive simple subgroup of a finite symmetric group is very rarely contained in a full wreath product in product action. All such simple permutation groups are determined in this paper. This remarkable conclusion is reached after a definition and detailed examination of ‘Cartesian decompositions’ of the permuted set, relating them to certain ‘Cartesian systems of subgroups’. These concepts, and the bijective connections between them, are explored in greater generality, with specific future applications in mind.

scholarly journals Finite generation of iterated wreath products in product action

2015 ◽  
Vol 105 (3) ◽  
pp. 205-214 ◽  
Author(s):  
Matteo Vannacci
Keyword(s):  
Wreath Products ◽  
Finite Generation ◽  
Product Action

scholarly journals The closures of wreath products in product action

2021 ◽  
Vol 60 (3) ◽  
pp. 286-297
Author(s):  
A. V. Vasilev ◽  
I. N. Ponomarenko
Keyword(s):  
Wreath Products ◽  
Product Action
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