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Cayley's Theorem

Artur Korniłowicz (2011)

Formalized Mathematics

The article formalizes the Cayley's theorem saying that every group G is isomorphic to a subgroup of the symmetric group on G.

Collineation group as a subgroup of the symmetric group

Fedor Bogomolov, Marat Rovinsky (2013)

Open Mathematics

Let ψ be the projectivization (i.e., the set of one-dimensional vector subspaces) of a vector space of dimension ≥ 3 over a field. Let H be a closed (in the pointwise convergence topology) subgroup of the permutation group 𝔖 ψ of the set ψ. Suppose that H contains the projective group and an arbitrary self-bijection of ψ transforming a triple of collinear points to a non-collinear triple. It is well known from [Kantor W.M., McDonough T.P., On the maximality of PSL(d+1,q), d ≥ 2, J. London Math. Soc.,...

Crossed product of cyclic groups

Ana-Loredana Agore, Dragoş Frățilă (2010)

Czechoslovak Mathematical Journal

All crossed products of two cyclic groups are explicitly described using generators and relations. A necessary and sufficient condition for an extension of a group by a group to be a cyclic group is given.

Every reasonably sized matrix group is a subgroup of S ∞

Robert Kallman (2000)

Fundamenta Mathematicae

Every reasonably sized matrix group has an injective homomorphism into the group S of all bijections of the natural numbers. However, not every reasonably sized simple group has an injective homomorphism into S .

Highly transitive subgroups of the symmetric group on the natural numbers

U. B. Darji, J. D. Mitchell (2008)

Colloquium Mathematicae

Highly transitive subgroups of the symmetric group on the natural numbers are studied using combinatorics and the Baire category method. In particular, elementary combinatorial arguments are used to prove that given any nonidentity permutation α on ℕ there is another permutation β on ℕ such that the subgroup generated by α and β is highly transitive. The Baire category method is used to prove that for certain types of permutation α there are many such possibilities for β. As a simple corollary,...

Invariance groups of finite functions and orbit equivalence of permutation groups

Eszter K. Horváth, Géza Makay, Reinhard Pöschel, Tamás Waldhauser (2015)

Open Mathematics

Which subgroups of the symmetric group Sn arise as invariance groups of n-variable functions defined on a k-element domain? It appears that the higher the difference n-k, the more difficult it is to answer this question. For k ≤ n, the answer is easy: all subgroups of Sn are invariance groups. We give a complete answer in the cases k = n-1 and k = n-2, and we also give a partial answer in the general case: we describe invariance groups when n is much larger than n-k. The proof utilizes Galois connections...

Maximal clones and maximal permutation groups

Péter P. Pálfy (2007)

Discussiones Mathematicae - General Algebra and Applications

A fundamental result in universal algebra is the theorem of Rosenberg describing the maximal subclones in the clone of all operations over a finite set. In group theory, the maximal subgroups of the symmetric groups are classified by the O'Nan-Scott Theorem. We shall explore the similarities and differences between these two analogous major results. In addition, we show that a primitive permutation group of diagonal type can be maximal in the symmetric group only if its socle is the direct product...

On B-injectors of symmetric groups Sₙ and alternating groups Aₙ: a new approach

M. I. AlAli, Bilal Al-Hasanat, I. Sarayreh, M. Kasassbeh, M. Shatnawi, A. Neumann (2009)

Colloquium Mathematicae

The aim of this paper is to introduce the notion of BG-injectors of finite groups and invoke this notion to determine the B-injectors of Sₙ and Aₙ and to prove that they are conjugate. This paper provides a new, more straightforward and constructive proof of a result of Bialostocki which determines the B-injectors of the symmetric and alternating groups.

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