The classification is here completed for the p-standard table algebras of order p3 for any rational prime p. These include the adjacency algebras of p-schemes of order p3. The remaining case, where ...the thin radical of the distinguished basis of the table algebra has order p2, is resolved. The algebras in this case are explicitly determined up to isomorphism as wreath products, partial wreath products, or as members of a more complex family called hexagonal standard table algebras. These are parametrized by certain correspondences introduced in this article called hexagonal functions, that are defined in general from a subset of a group to an arbitrary set.
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The properties and activities of chemicals are strongly related to their molecular structures. Topological indices defined on these molecular structures are capable to predict those properties and ...activities. In this article, a new topological index named as neighborhood Zagreb index (MN) is presented. Here the chemical importance of the MN index is investigated and it is shown that the newly introduced index is useful in predicting physico-chemical properties with high accuracy compared to some well-established and often used indices. The isomer-discrimination ability of MN is also examined. To demonstrate how the computational formula of the novel index for chemical compounds is simple and convenient, the chemical structures of favipiravir and hydroxychloroquine are used. In addition, some explicit results for this index of different product graphs such as Cartesian, tensor and wreath product are derived. Some of these results are applied to obtain the MN index of some special structures.
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We study the base sizes of finite quasiprimitive permutation groups of twisted wreath type, which are precisely the finite permutation groups with a unique minimal normal subgroup that is also ...non-abelian, non-simple and regular. Every permutation group of twisted wreath type is permutation isomorphic to a twisted wreath product G=Tk:P acting on its base group Ω=Tk, where T is some non-abelian simple group and P is some group acting transitively on k={1,…,k} with k⩾2. We prove that if G is primitive on Ω and P is quasiprimitive on k, then G has base size 2. We also prove that the proportion of pairs of points that are bases for G tends to 1 as |G|→∞ when G is primitive on Ω and P is primitive on k. Lastly, we determine the base size of any quasiprimitive group of twisted wreath type up to four possible values (and three in the primitive case). In particular, we demonstrate that there are many families of primitive groups of twisted wreath type with arbitrarily large base sizes.
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Drimbe and Vaes proved an orbit equivalence superrigidity theorem for left-right wreath product actions in the measurable setting. We establish the counterpart result in the topological setting for ...continuous orbit equivalence. This gives us minimal, topologically free actions that are continuous orbit equivalence superrigid. One main ingredient for the proof is to show continuous cocycle superrigidity for certain generalized full shifts, extending our previous result with Chung.
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We describe bases for the morphism spaces of the Frobenius Heisenberg categories associated to a symmetric graded Frobenius algebra, proving several open conjectures. Our proof uses a categorical ...comultiplication and generalized cyclotomic quotients of the category. We use our basis theorem to prove that the Grothendieck ring of the Karoubi envelope of the Frobenius Heisenberg category recovers the lattice Heisenberg algebra associated to the Frobenius algebra.
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In this paper we explore a generic notion of superrigidity for von Neumann algebras L(G) and reduced C⁎-algebras Cr⁎(G) associated with countable discrete groups G. This allows us to classify these ...algebras for various new classes of groups G from the realm of coinduced groups.
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Given a group G and
n
≥
0
, let W(G, n) be the associated iterated wreath product-unrestricted when G is infinite-viewed as a permutation group on G
n
. We prove that the normalizer of W(G, n) in the ...symmetric group
S
(
G
n
)
is equal to
M
n
⋉
W
(
G
,
n
)
, where M
n
is isomorphic to
Aut
(
G
)
n
. The action of
Aut
(
G
)
n
on W(G, n) is recursively described.
Communicated by Mark Lewis
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