For a finite group G , the prime graph Γ ( G ) (also known as Gruenberg-Kegel graph) is defined to be the graph where the vertices are the primes that divide | G | such that two vertices p and q share an edge if and only if there is an element of order pq in G . The prime graphs of solvable groups have been classified. The prime graphs of groups whose noncyclic composition factors are isomorphic to a single nonabelian simple group T where | T | is divisible by three or four distinct primes have been classified except for the cases where T = PSL ( 2 , q ) for q ≠ 2 5 and | PSL ( 2 , q ) | is divisible by exactly four primes. In this paper, we provide criteria for general classification results for certain classes of T , and then use them to classify the prime graphs of some T -solvable groups for T a suitably small PSL ( 2 , q ) -group. We also provide general results on the prime graphs of T -solvable groups where T is a member of the possibly infinite family of groups PSL ( 2 , 2 f ) such that f ≥ 5 , f is prime, and | PSL ( 2 , 2 f ) | is divisible by exactly four primes. This is the first paper to prove general results about the prime graphs of T -solvable groups where T belongs to a large (probably infinite) family of groups.
A path Pn is a graph with n vertices vi, i=1,…,n and edges {vi,vi+1}, i=1,…,n−1. If we replace each vertex with a graph Gi which is either a complete or an empty graph, and have edges between all vertices of Gi and Gi+1, we have a path join graph. In this article, we are interested in 2-linear resolution of the edge rings of path join graphs. We completely characterize path join graphs with 2-linear resolution along with their Hilbert series and Betti numbers. Also, we present combinatorial formulae for the regularity (Castelnuovo-Mumford) and projective dimension of some path join graphs. Finally, we do similar investigations for cycle graphs.
In 1986, Elkies disproved Euler's conjecture for fourth powers by demonstrating that the equationA4+B4+C4=D4 has infinitely many integer solutions, using a method based on hyperelliptic curves. The first explicit solution is26824404+153656394+187967604=206156734, but subsequent solutions grow extremely large. After learning of Elkies' result, Frye performed an exhaustive computer search and found the smallest solution:958004+2175194+4145604=4224814. When the search is restricted to primitive solutions whose values do not exceed 30 digits, twenty seven such solutions are presently known.In this paper, we build upon Elkies' method to obtain thirteen new primitive solutions, with values reaching up to 30 digits. Our analysis focuses on two hyperelliptic curves. The first curve gives thirteen solutions: six previously known and seven new. This group of solutions is derived from the three base solutions of Frye, MacLeod, and Tomita. The second curve gives fourteen solutions: eight known and six new. These are all generated by the three base solutions of Elkies, MacLeod, and Tomita.
For a fixed integer r≥2, we prove that the compressed primitivity problem in the free group Fr=F(x1,…,xr) is decidable in nondeterministic polynomial time. That is, for a straight-line program A over {x1,…,xr}±1 representing an element g∈Fr, the problem of deciding whether g is primitive in Fr belongs to NP, with input measured by the size of A. For r=2, we prove that this problem is decidable in deterministic polynomial time. We also show that, in every fixed rank r≥2, automorphic minimality of the conjugacy class of a compressed word in Fr is decidable in deterministic polynomial time.
A quasigroup (Q,⋅) is called maximally nonassociative if for all (x,y,z)∈Q3, the equality (x⋅y)⋅z=x⋅(y⋅z) implies that x=y=z. We follow up on the 2020 article “Maximal nonassociativity via nearfields” by Drápal and Lisoněk, in which the authors first proved the existence of maximally nonassociative quasigroups of certain orders. They studied quasigroups derived from quadratic nearfields, and based on computational results, conjectured that the asymptotic density of maximally nonassociative quasigroups among them lies between 0.288 and 0.29. Using the Weil bound as our main tool, we confirm their conjecture and show that the asymptotic density equals 37/128=0.2890625.
