We prove two results. (1) There is an absolute constant D such that for any finite quasisimple group S, given 2D arbitrary automorphisms of S, every element of S is equal to a product of D ‘twisted commutators’ defined by the given automorphisms. (2) Given a natural number q, there exist C = C(q) and M = M(q) such that: if S is a finite quasisimple group with |S/Z(S)| > C, βj (j = 1, . . . ,M) ...

It is already shown that a Boolean function for a NP-complete problem can be computed by a polynomial-sized circuit if its variables have enough number of automorphisms. Looking at this previous study from the different perspective gives us the idea that the small number of automorphisms might be a barrier for a polynomial time solution for NP-complete problems. Here I show that by interpreting...

A class of structures C is said to have the extension property for partial automorphisms (EPPA) if, whenever C 1 and C 2 are structures in C, C 1 nite, C 1 C 2 , and p 1 , p 2 , : : :, p n are partial automorphisms of C 1 extending to automorphisms of C 2 , then there exist a nite structure C 3 in C and automorphisms 1 , 2 , : : :, n of C 3 extending the p i. We will prove that some classes of ...

The aim of this work is to introduce an approach for interval-valued R-implications, which satisfy some analogous properties of R-implications. We show that the best interval representation of an R-implication that is obtained from a left continuous tnorm coincides with the interval-valued R-implication obtained from the best interval representation of such t-norm, whenever this is an inclusion...

This paper establishes decomposability for composition operators induced on the Hardy spaces H (1 ≤ p < ∞) by parabolic linear fractional self-maps of the unit disc that are not automorphisms. This result, along with a recent theorem of Miller and Miller [15], shows that no such composition operator is supercyclic. The work here completes part of a previous investigation [1] where the author an...

The transition graph of a cellular automaton (CA) represents the CA’s global dynamics. The automorphisms of the transition graph are its self-isomorphisms or “symmetries”, and in a sense these are precisely the symmetries of the CA’s dynamics. Previously we have argued that studying how the total number of automorphisms varies with the number of cells on which the CA operates yields a partial c...

A class of structures C is said to have the extension property for partial automorphisms (EPPA) if, whenever C1 and C2 are structures in C, C1 finite, C1 ⊆ C2, and p1, p2, . . . , pn are partial automorphisms of C1 extending to automorphisms of C2, then there exist a finite structure C3 in C and automorphisms α1, α2, . . . , αn of C3 extending the pi. We will prove that some classes of structur...

The automorphisms of free groups with boundaries form a family of groups An,k closely related to mapping class groups, with the standard automorphisms of free groups as An,0 and (essentially) the symmetric automorphisms of free groups as A0,k . We construct a contractible space Ln,k on which An,k acts with finite stabilizers and finite quotient space and deduce a range for the virtual cohomolog...

Let $\mathcal{C}$ be a smooth, projective, genus $g\geq 2$ curve, defined over $\mathbb{C}$. Then has \emph{many automorphisms} if its corresponding moduli point $p \in \mathcal{M}_g$ neighborhood $U$ in the complex topology, such that all curves to points $U \setminus \{p \}$ have strictly fewer automorphisms than $\mathcal{C}$. We compute completely list of superelliptic having many automorph...

Recently, Shestakov-Umirbaev solved Nagata’s conjecture on an automorphism of a polynomial ring. To solve the conjecture, they defined notions called reductions of types I–IV for automorphisms of a polynomial ring. An automorphism admitting a reduction of type I was first found by Shestakov-Umirbaev. Using a computer, van den Essen–Makar-Limanov–Willems gave a family of such automorphisms. In t...