The present paper records more details of the relationship between primitive elements and palindromes in F2, the free group of rank two. We characterize the conjugacy classes of palindromic primitive elements as those in which cyclically reduced words have odd length. We identify large palindromic subwords of certain primitives in conjugacy classes which contain cyclically reduced words of even...

Abstract We construct via Fraïssé amalgamation an ????-categorical structure whose automorphism group is infinite oligomorphic Jordan primitive permutation preserving a “limit of ????-relations”. The construction based on semilinear order elements are labelled by sets carrying ????-relation, with strong coherence conditions governing how these ????-sets inter-related.

We give a new characterization of primitive permutation groups tied to the notion completely reachable automata. Also, we introduce sync-maximal state complexity set synchronizing words certain associated automata and show that they are contained between 2-homogeneous groups. Lastly, define k-reachable in analogy with motivated by our But results group degree n \(6 \le k - 6\) is either alterna...

In this paper, we study two-dimensional divisor problems of the Fourier coefficients some automorphic product L-functions attached to primitive holomorphic cusp form f(z) with weight k for full modular group SL2(Z). Additionally, establish upper bound and asymptotic formula these on average, respectively.

In this paper, we study the average behaviour of coefficients triple product $L$-functions and some related corresponding to normalized primitive holomorphic cusp form $f(z)$ weight $k$ for full modular group $SL(2,\mathbbZ).$ Here call a if it is an eigenfunction all Hecke operators simultaneously.

Almost all primitive permutation groups of degree n have order at most n · Q[log2 n]−1 i=0 (n − 2) < n2 , or have socle isomorphic to a direct power of some alternating group. The Mathieu groups, M11, M12, M23 and M24 are the four exceptions. As a corollary the sharp version of a theorem of Praeger and Saxl is established, where M12 turns out to be the ”largest” primitive group. For an applicat...

We prove that, for a primitive permutation group G acting on a set X of size n, other than the alternating group, the probability that Aut(X, Y ) = G for a random subset Y of X, tends to 1 as n → ∞. So the property of the title holds for all primitive groups except the alternating groups and finitely many others. This answers a question of M. H. Klin. Moreover, we give an upper bound n for the ...

Let $G$ be a transitive permutation group on finite set $\Omega$ and recall that base for is subset of with trivial pointwise stabiliser. The size $G$, denoted $b(G)$, the minimal base. If $b(G)=2$ then we can study Saxl graph $\Sigma(G)$ which has vertex two vertices are adjacent if they form This vertex-transitive graph, conjectured to connected diameter at most $2$ when primitive. In this pa...