Closed sets of finitary functions between products of finite fields of coprime order

We investigate the finitary functions from a finite product of finite fields $$\prod _{j =1}^m\mathbb {F}_{q_j} = {\mathbb K}$$ ∏ j = 1 m F q j = K to a finite product of finite fields $$\prod _{i =1}^n\mathbb {F}_{p_i} = {\mathbb {F}}$$ ∏ i = 1 n F p i = F , where $$|{\mathbb K}|$$ | K | and $$|{\mathbb {F}}|$$ | F | are coprime. An $$({\mathbb {F}},{\mathbb K})$$ ( F , K ) -linearly closed clonoid is a subset of these functions which is closed under composition from the right and from the left with linear mappings. We give a characterization of these subsets of functions through the $${\mathbb {F}}_p[{\mathbb K}^{\times }]$$ F p [ K × ] -submodules of $$\mathbb {F}_p^{{\mathbb K}}$$ F p K , where $${\mathbb K}^{\times }$$ K × is the multiplicative monoid of $${\mathbb K}= \prod _{i=1}^m {\mathbb {F}}_{q_i}$$ K = ∏ i = 1 m F q i . Furthermore we prove that each of these subsets of functions is generated by a set of unary functions and we provide an upper bound for the number of distinct $$({\mathbb {F}},{\mathbb K})$$ ( F , K ) -linearly closed clonoids.

Expansions of abelian square-free groups

We investigate finitary functions from [Formula: see text] to [Formula: see text] for a square-free number [Formula: see text]. We show that the lattice of all clones on the square-free set [Formula: see text] which contain the addition of [Formula: see text] is finite. We provide an upper bound for the cardinality of this lattice through an injective function to the direct product of the lattices of all [Formula: see text]-linearly closed clonoids, [Formula: see text], to the [Formula: see text] power, where [Formula: see text]. These lattices are studied in [S. Fioravanti, Closed sets of finitary functions between products of finite fields of pair-wise coprime order, preprint (2020), arXiv:2009.02237 ] and there we can find an upper bound for their cardinality. Furthermore, we prove that these clones can be generated by a set of functions of arity at most [Formula: see text].

Closed sets of finitary functions between finite fields of coprime order

We investigate the finitary functions from a finite field $$\mathbb {F}_q$$ F q to the finite field $$\mathbb {F}_p$$ F p , where p and q are powers of different primes. An $$(\mathbb {F}_p,\mathbb {F}_q)$$ ( F p , F q ) -linearly closed clonoid is a subset of these functions which is closed under composition from the right and from the left with linear mappings. We give a characterization of these subsets of functions through the invariant subspaces of the vector space $$\mathbb {F}_p^{\mathbb {F}_q\backslash \{0\}}$$ F p F q \ { 0 } with respect to a certain linear transformation with minimal polynomial $$x^{q-1} - 1$$ x q - 1 - 1 . Furthermore we prove that each of these subsets of functions is generated by one unary function.

Improved Upper Bounds for the Order of Some Classes of Abelian Cayley and Circulant Graphs of Diameter Two

In the degree-diameter problem for Abelian Cayley and circulant graphs of diameter 2 and arbitrary degree d there is a wide gap between the best lower and upper bounds valid for all d, being quadratic functions with leading coefficient 1/4 and 1/2 respectively. Recent papers have presented constructions which increase the coefficient of the lower bound to be at or just below 3/8 for sparse sets of degree d related to primes of specific congruence classes. The constructions use the direct product of the multiplicative and additive subgroups of a Galois field and a specific cyclic group of coprime order. It was anticipated that this approach would be capable of yielding further improvement towards the upper bound value of 1/2. In this paper, however, it is proved that the quadratic coefficient of the order of families of Abelian Cayley graphs of this class of construction can never exceed the value of 3/8, establishing an asymptotic limit of 3/8 for the quadratic coefficient of families of extremal graphs of this class. By applying recent results from number theory these constructions can be extended to be valid for every degree above some threshold, establishing an improved asymptotic lower bound approaching 3/8 for general Abelian Cayley and circulant graphs of diameter 2 and arbitrary degree d.

A characterization of nilpotent Moufang loops of odd order

Glauberman and Wright in [G. G. Glauberman and C. R. B. Wright, Nilpotence of finite Moufang 2-loops, J. Algebra 8 (1968) 415–417] proved that a nilpotent Moufang loop is the direct product of [Formula: see text]-loops for some primes [Formula: see text], consequently the elements of coprime order commute in a nilpotent Moufang loop. In this paper, we prove that in Moufang loops of odd order this condition is equivalent to the central nilpotence.

Action of a Frobenius-like group with kernel having central derived subgroup

A finite group [Formula: see text] is said to be Frobenius-like if it has a nontrivial nilpotent normal subgroup [Formula: see text] with a nontrivial complement [Formula: see text] such that [Formula: see text] for all nonidentity elements [Formula: see text]. Suppose that a finite group [Formula: see text] admits a Frobenius-like group of automorphisms [Formula: see text] of coprime order with [Formula: see text] In case where [Formula: see text] we prove that the groups [Formula: see text] and [Formula: see text] have the same nilpotent length under certain additional assumptions.

A GENERALIZED FIXED-POINT-FREE ACTION

In this paper we study the structure of a finite group G admitting a solvable group A of automorphisms of coprime order so that for any x ∈ CG(A) of prime order or of order 4, every conjugate of x in G is also contained in CG(A). Under this hypothesis it is proven that the subgroup [G, A] is solvable. Also an upper bound for the nilpotent height of [G, A] in terms of the number of primes dividing the order of A is obtained in the case where A is abelian.

On the Wielandt length of a finite supersoluble group

We develop a technique for calculating the Wielandt subgroup of a semidirect product of two finite groups of coprime order. We apply this technique to calculate the Wielandt length of a supersoluble group in terms of the Wielandt lengths of its Sylow subgroups (for small Wielandt lengths) and in terms of the nilpotency classes of its Sylow subgroups.