Random Cyclic Triangle-Free Graphs of Prime Order

Cyclic triangle-free process (CTFP) is the cyclic analog of the triangle-free process. It begins with an empty graph of order n and generates a cyclic graph of order n by iteratively adding parameters, chosen uniformly at random, subject to the constraint that no triangle is formed in the cyclic graph obtained, until no more parameters can be added. The structure of a cyclic triangle-free graph of the prime order is different from that of composite integer order. Cyclic graphs of prime order have better properties than those of composite number order, which enables generating cyclic triangle-free graphs more efficiently. In this paper, a novel approach to generating cyclic triangle-free graphs of prime order is proposed. Based on the cyclic graphs of prime order, obtained by the CTFP and its variant, many new lower bounds on R 3 , t are computed, including R 3,34 ≥ 230 , R 3,35 ≥ 242 , R 3,36 ≥ 252 , R 3,37 ≥ 264 , R 3,38 ≥ 272 . Our experimental results demonstrate that all those related best known lower bounds, except the bound on R 3,34 , are improved by 5 or more.

Diagonal groups and arcs over groups

In an earlier paper by three of the present authors and Csaba Schneider, it was shown that, for $$m\ge 2$$ m ≥ 2 , a set of $$m+1$$ m + 1 partitions of a set $$\Omega $$ Ω , any m of which are the minimal non-trivial elements of a Cartesian lattice, either form a Latin square (if $$m=2$$ m = 2 ), or generate a join-semilattice of dimension m associated with a diagonal group over a base group G. In this paper we investigate what happens if we have $$m+r$$ m + r partitions with $$r\ge 2$$ r ≥ 2 , any m of which are minimal elements of a Cartesian lattice. If $$m=2$$ m = 2 , this is just a set of mutually orthogonal Latin squares. We consider the case where all these squares are isotopic to Cayley tables of groups, and give an example to show the groups need not be all isomorphic. For $$m>2$$ m > 2 , things are more restricted. Any $$m+1$$ m + 1 of the partitions generate a join-semilattice admitting a diagonal group over a group G. It may be that the groups are all isomorphic, though we cannot prove this. Under an extra hypothesis, we show that G must be abelian and must have three fixed-point-free automorphisms whose product is the identity. (We describe explicitly all abelian groups having such automorphisms.) Under this hypothesis, the structure gives an orthogonal array, and conversely in some cases. If the group is cyclic of prime order p, then the structure corresponds exactly to an arc of cardinality $$m+r$$ m + r in the $$(m-1)$$ ( m - 1 ) -dimensional projective space over the field with p elements, so all known results about arcs are applicable. More generally, arcs over a finite field of order q give examples where G is the elementary abelian group of order q. These examples can be lifted to non-elementary abelian groups using p-adic techniques.

Profinite genus of the fundamental groups of compact flat manifolds with holonomy group of prime order

In this article, we calculate the profinite genus of the fundamental group of an 𝑛-dimensional compact flat manifold 𝑋 with holonomy group of prime order. As consequence, we prove that if n ⩽ 21 n\leqslant 21 , then 𝑋 is determined among all 𝑛-dimensional compact flat manifolds by the profinite completion of its fundamental group. Furthermore, we characterize the isomorphism class of the profinite completion of the fundamental group of 𝑋 in terms of the representation genus of its holonomy group.

Profinite groups with an automorphism of prime order whose fixed points have finite Engel sinks

A right Engel sink of an element g of a group G is a set $${{\mathscr {R}}}(g)$$ R ( g ) such that for every $$x\in G$$ x ∈ G all sufficiently long commutators $$[...[[g,x],x],\dots ,x]$$ [ . . . [ [ g , x ] , x ] , ⋯ , x ] belong to $${\mathscr {R}}(g)$$ R ( g ) . (Thus, g is a right Engel element precisely when we can choose $${{\mathscr {R}}}(g)=\{ 1\}$$ R ( g ) = { 1 } .) We prove that if a profinite group G admits a coprime automorphism $$\varphi $$ φ of prime order such that every fixed point of $$\varphi $$ φ has a finite right Engel sink, then G has an open locally nilpotent subgroup. A left Engel sink of an element g of a group G is a set $${{\mathscr {E}}}(g)$$ E ( g ) such that for every $$x\in G$$ x ∈ G all sufficiently long commutators $$[...[[x,g],g],\dots ,g]$$ [ . . . [ [ x , g ] , g ] , ⋯ , g ] belong to $${{\mathscr {E}}}(g)$$ E ( g ) . (Thus, g is a left Engel element precisely when we can choose $${\mathscr {E}}(g)=\{ 1\}$$ E ( g ) = { 1 } .) We prove that if a profinite group G admits a coprime automorphism $$\varphi $$ φ of prime order such that every fixed point of $$\varphi $$ φ has a finite left Engel sink, then G has an open pronilpotent-by-nilpotent subgroup.

A Novel Provably Secure Key-Agreement Using Secret Subgroup Generator

In this paper, a new key-agreement scheme is proposed and analyzed. In addition to being provably secure in the shared secret key indistinguishability model under Decisional Diffie-Hellman assumption for subgroup of matrices over GF(2) with prime order, which considered as basic security requirement, the scheme has an interesting feature; it uses exponentiations over cyclic group using hidden secret subgroup generator as a platform for the key exchange, whereby - unlike many other exponentiation based key exchange schemes - it transcends the reliance on intractability of Discrete Logarithm Problem in its security.

On Finite Groups with an Automorphism of Prime Order Whose Fixed Points Have Bounded Engel Sinks

A left Engel sink of an elementgof a groupGis a set$${\mathscr {E}}(g)$$E(g)such that for every$$x\in G$$x∈Gall sufficiently long commutators$$[...[[x,g],g],\dots ,g]$$[...[[x,g],g],⋯,g]belong to$${\mathscr {E}}(g)$$E(g). (Thus,gis a left Engel element precisely when we can choose$${\mathscr {E}}(g)=\{ 1\}$$E(g)={1}.) We prove that if a finite groupGadmits an automorphism$$\varphi $$φof prime order coprime to |G| such that for some positive integermevery element of the centralizer$$C_G(\varphi )$$CG(φ)has a left Engel sink of cardinality at mostm, then the index of the second Fitting subgroup$$F_2(G)$$F2(G)is bounded in terms ofm. A right Engel sink of an elementgof a groupGis a set$${\mathscr {R}}(g)$$R(g)such that for every$$x\in G$$x∈Gall sufficiently long commutators$$[\ldots [[g,x],x],\dots ,x]$$[…[[g,x],x],⋯,x]belong to$${\mathscr {R}}(g)$$R(g). (Thus,gis a right Engel element precisely when we can choose$${\mathscr {R}}(g)=\{ 1\}$$R(g)={1}.) We prove that if a finite groupGadmits an automorphism$$\varphi $$φof prime order coprime to |G| such that for some positive integermevery element of the centralizer$$C_G(\varphi )$$CG(φ)has a right Engel sink of cardinality at mostm, then the index of the Fitting subgroup$$F_1(G)$$F1(G)is bounded in terms ofm.