Prime Order Research Articles

Prime order element graph of a group

Prime order element graph of a group

Flag-transitive point-primitive symmetric 2-designs of prime square order

Flag-transitive point-primitive symmetric 2-designs of prime square order

On Hall normally embedded subgroups and the p-nilpotency of finite groups

Let G be a finite group. A subgroup H of G is called Hall normally embedded in G if H is a Hall subgroup of the normal closure H G of H in G. In this paper, we shall obtain some characterizations about p-nilpotency of G under the assumption that certain some subgroups of G of prime power order are Hall normally embedded in G.

Spectral Pseudorandomness and the Road to Improved Clique Number Bounds for Paley Graphs

We study subgraphs of Paley graphs of prime order p induced on the sets of vertices extending a given independent set of size a to a larger independent set. Using a sufficient condition due to Kunisky (2023), we show that estimates on a new family of necklace character sums would imply that, as p → ∞ , the empirical spectral distributions of the adjacency matrices of any sequence of such subgraphs have the same weak limit (after rescaling) as those of subgraphs induced on a random set including each vertex independently with probability 2 − a , namely, a Kesten-McKay law with parameter 2 a . We prove the necessary estimates for a = 1, obtaining in the process an alternate proof of a character sum equidistribution result of Xi (2022), and provide numerical evidence for this weak convergence for a ≥ 2 . We also conjecture that the minimum eigenvalue of any such sequence converges (after rescaling) to the left edge of the corresponding Kesten-McKay law, and provide numerical evidence for this convergence. Finally, we show that, once a ≥ 3 , this (conjectural) convergence of the minimum eigenvalue would imply bounds on the clique number of the Paley graph improving on the current state of the art due to Hanson and Petridis (2021), and that this convergence for all a ≥ 1 would imply that the clique number is o ( p ) .

Molecular Topology Index of a Zero Divisor Graph on a Ring of Integers Modulo Prime Power Order

In chemistry, graph theory has been widely utilized to address molecular problems, with numerous applications in graph theory and ring theory within this field. One of these applications involves topological indices that represent chemical structures with numerical values. Various types of topological indices exist, including the Wiener index, the first Zagreb index, and the hyper-Wiener index. In the context of this research, the values of the Wiener index, the first Zagreb index, and the hyper-Wiener index for zero-divisor graphs on the ring of integers modulo a prime power order will be explored through a literature review and conjecture.

The structure of finite groups whose elements outside a normal subgroup have prime power orders

The structure of groups in which every element has prime power order (CP-groups) is extensively studied. We first investigate the properties of group $G$ such that each element of $G\setminus N$ has prime power order. It is proved that $N$ is solvable or every non-solvable chief factor $H/K$ of $G$ satisfying $H\leq N$ is isomorphic to $PSL_2(3^f)$ with $f$ a 2-power. This partially answers the question proposed by Lewis in 2023, asking whether $G\cong M_{10}$ ? Furthermore, we prove that if each element $x\in G\backslash N$ has prime power order and ${\bf C}_G(x)$ is maximal in $G$ , then $N$ is solvable. Relying on this, we give the structure of group $G$ with normal subgroup $N$ such that ${\bf C}_G(x)$ is maximal in $G$ for any element $x\in G\setminus N$ . Finally, we investigate the structure of a normal subgroup $N$ when the centralizer ${\bf C}_G(x)$ is maximal in $G$ for any element $x\in N\setminus {\bf Z}(N)$ , which is a generalization of results of Zhao, Chen, and Guo in 2020, investigating a special case that $N=G$ for our main result. We also provide a new proof for Zhao, Chen, and Guo's results above.

Some New Results on S-prime Ideals of a Finite Commutative Ring as S-meet Semilattice

Let ℜ be the finite commutative ring with unity and <i>I<sub>s</sub></i> be the S-prime ideal of a ring ℜ. The set <i>L<sub>s</sub></i> forms a partially ordered set (poset) by the subset relation. Initially, the interplay of the semilattice theoretic properties of a poset with the ring theoretic properties are studied with suitable examples. The number of maximal chain of a poset is compared with the number of prime ideals of a ring. It is proved that every maximal element of a poset is the prime ideal of a ring. A prime order ring is shown as a lattice. If the order of the ring is the product of two primes, then the trivial ideal is expressed as the meet of every pair of a poset. Further, the cardinality of the poset is determined in terms of the divisors of the order of the ring . A new meet-semilattice called the S-meet semilattice (<i>L<sub>s</sub></i>, ⋀, ⊆) is defined and the generalized Hasse diagrams of the S-meet semilattice of a ring of prime powers, product of prime powers are drawn in this paper in order to find the properties of S-meet semilattice. Finally, the ideals, the prime ideals and the maximal ideals of the S-meet semilattice are described in terms of the down-sets of S-meet semilattice where the results are listed with an example at the end.

On parallelisms of PG(3,4) with automorphisms of order 2

Let PG(n,q) be the n-dimensional projective space over the finite field Fq. A spread in PG(n,q) is a set of mutually skew lines which partition the point set. A parallelism is a partition of the set of lines by spreads. The classification of parallelisms in small finite projective spaces is of interest for problems from projective geometry, design theory, network coding, error-correcting codes, and cryptography. All parallelisms of PG(3,2) and PG(3,3) are known. Parallelisms of PG(3,4) which are invariant under automorphisms of odd prime orders have also been classified. The present paper contributes to the classification of parallelisms in PG(3,4) with automorphisms of even order. We focus on cyclic groups of order four and the group of order two generated by a Baer involution. We examine invariants of the parallelisms such as the full automorphism group, the type of spreads and questions of duality. The results given in this paper show that the number of parallelisms of PG(3,4) is at least 8675365. Some future directions of research are outlined.

Tensor product and quandle rings of connected quandles of prime order

Let C be field of complex numbers and X be a connected quandle of prime order. We study the regular representation of X by describing the quandle ring C [ X ] as direct sum of right simple ideals. We provide description of tensor product of connected quandles of prime order. We further discuss multiplicity freeness of quandle ring decomposition for connected quandles of order ≤ 47 and prove that C [ X ] decomposes multiplicity free for affine connected quandle X.

The quasiprimitive almost elusive groups

Let [Formula: see text] be a nontrivial transitive permutation group on a finite set [Formula: see text] and recall that an element of [Formula: see text] is a derangement if it has no fixed points. Derangements always exist by a classical theorem of Jordan, but there are so-called elusive groups that do not contain any derangements of prime order. In a recent paper, Burness and the author introduced the family of almost elusive groups, which contain a unique conjugacy class of derangements of prime order. In this paper, we complete the classification of the quasiprimitive almost elusive groups.

Finite supersolvable groups and Hall normally embedded subgroups of prime power order

Finite supersolvable groups and Hall normally embedded subgroups of prime power order

Computational Enumeration of André Planes

In this paper, we address computational questions surrounding the enumeration of non-isomorphic André planes for any prime power order q . We are particularly focused on providing a complete enumeration of all such planes for relatively small orders (up to 125), as well as developing computationally efficient ways to count the number of isomorphism classes for other orders where enumeration is infeasible. André planes of all dimensions over their kernel are considered.

Finite groups in which every commutator has prime power order

Finite groups in which every element has prime power order (EPPO-groups) are nowadays fairly well understood. For instance, if G is a soluble EPPO-group, then the Fitting height of G is at most 3 and |π(G)|⩽2 (Higman, 1957). Moreover, Suzuki showed that if G is insoluble, then the soluble radical of G is a 2-group and there are exactly eight nonabelian simple EPPO-groups.In the present work we concentrate on finite groups in which every commutator has prime power order (CPPO-groups). Roughly, we show that if G is a CPPO-group, then the structure of G′ is similar to that of an EPPO-group. In particular, we show that the Fitting height of a soluble CPPO-group is at most 3 and |π(G′)|⩽3. Moreover, if G is insoluble, then R(G′) is a 2-group and G′/R(G′) is isomorphic to a simple EPPO-group.

Galois subcovers of the Hermitian curve in characteristic p with respect to subgroups of order p2

A (projective, geometrically irreducible, non-singular) curve X defined over a finite field Fq2 is maximal if the number Nq2 of its Fq2-rational points attains the Hasse-Weil upper bound, that is Nq2=q2+2gq+1 where g is the genus of X. An important question, also motivated by applications to algebraic-geometry codes, is to find explicit equations for maximal curves. For a few curves which are Galois covered of the Hermitian curve, this has been done so far ad hoc, in particular in the cases where the Galois group has prime order. In this paper we obtain explicit equations of all Galois covers of the Hermitian curve with Galois group of order p2 where p is the characteristic of Fq2. Doing so we also determine the Fq2-isomorphism classes of such curves and describe their full Fq2-automorphism groups.

Symmetric 2‐(36,15,6) $(36,15,6)$ designs with an automorphism of order two

AbstractBouyukliev, Fack and Winne classified all 2‐ designs that admit an automorphism of odd prime order, and gave a partial classification of such designs that admit an automorphism of order 2. In this paper, we give the classification of all symmetric 2‐ designs that admit an automorphism of order two. It is shown that there are exactly nonisomorphic such designs, of which are self‐dual designs. The ternary linear codes spanned by the incidence matrices of these designs are computed. Among these codes, there are near‐extremal self‐dual codes with previously unknown weight distributions.

Brieskorn spheres, cyclic group actions and the Milnor conjecture

AbstractIn this paper we further develop the theory of equivariant Seiberg–Witten–Floer cohomology of the two authors, with an emphasis on Brieskorn homology spheres. We obtain a number of applications. First, we show that the knot concordance invariants defined by the first author satisfy for torus knots, whenever is a prime not dividing . Since is a lower bound for the slice genus, this gives a new proof of the Milnor conjecture. Second, we prove that a free cyclic group action on a Brieskorn homology 3‐sphere does not extend smoothly to any homology 4‐ball bounding . In the case of a non‐free cyclic group action of prime order, we prove that if the rank of is greater than times the rank of , then the ‐action on does not extend smoothly to any homology 4‐ball bounding . Third, we prove that for all but finitely many primes a similar non‐extension result holds in the case that the bounding 4‐manifold has positive‐definite intersection form. Finally, we also prove non‐extension results for equivariant connected sums of Brieskorn homology spheres.

Verifiable Encryption from MPC-in-the-Head

Verifiable encryption (VE) is a protocol where one can provide assurance that an encrypted plaintext satisfies certain properties, or relations. It is an important building block in cryptography with many useful applications, such as key escrow, group signatures, optimistic fair exchange, and others. However, the majority of previous VE schemes are restricted to instantiation with specific public-key encryption schemes or relations. In this work, we propose a novel framework that realizes VE protocols using zero-knowledge proof systems based on the MPC-in-the-head paradigm (Ishai et al. STOC 2007). Our generic compiler can turn a large class of zero-knowledge proofs into secure VE protocols for any secure public-key encryption scheme with the undeniability property, a notion that essentially guarantees binding of encryption when used as a commitment scheme. Our framework is versatile: because the circuit proven by the MPC-in-the-head prover is decoupled from a complex encryption function, the work of the prover is focused on proving the encrypted data satisfies the relation, not the proof of plaintext knowledge. Hence, our approach allows for instantiation with various combinations of properties about the encrypted data and encryption functions. We then consider concrete applications, to demonstrate the efficiency of our framework, by first giving a new approach and implementation to verifiably encrypt discrete logarithms in any prime order group more efficiently than was previously known. Then we give the first practical verifiable encryption scheme for AES keys with post-quantum security, along with an implementation and benchmarks.

Non-Compatible Action Graph and Its Adjacency Matrix for The Non-abelian Tensor Product for Groups of Prime Power Order

This article focused on the notion of the non-abelian tensor product of groups of prime power order. Particularly, it presented new graph named as Non-compatible action graph and discussed some of its properties. Moreover, this graph concentrated on the case of non-compatible actions of the tensor product of two finite -groups. Furthermore, its adjacency matrix has been determined and discussed in detail. Moreover, the adjacency matrix has been denoted by A(G) and its inputs are 1 whenever there is adjacency and 0 otherwise.

Indeks Padmakar-Ivan dan indeks Randic pada graf non-koprima dari grup bilangan bulat modulo

Graph theory, introduced by the Swiss mathematician Leonhard Euler in 1736, has played a pivotal role in solving real-world problems since its inception, notably exemplified by Euler's solution to the Konigsberg Bridge problem. Its applications extend to various domains, including scheduling, shortest path routing, and chemical structure representation. In chemistry, graphs are extensively used to depict molecular structures and chemical compounds, aiding in visualizing atomic connections and overall compound configurations. Topology indices, such as the Padmakar-Ivan (PI) and Randic indices, provide numerical values capturing chemical bonding relationships. Beyond chemical structures, these indices find applications in abstract algebraic graph representations. Recent research, exemplified by Husni et al.'s work on the harmonic and Gutman indices, explores these indices in coprime graphs of integer groups modulo prime power orders. Additionally, studies on non-coprime graphs of integer groups modulo reveal unique characteristics and invariants, shedding light on their structure. The non-coprime graph is a graph with two vertices said to be adjacent if the greatest common divisor (GCD) of their orders is not equal to one. This paper aims to investigate the topological indices, specifically the Padmakar-Ivan and Randic indices, in non-coprime graphs of integer groups modulo, adding depth to our understanding of their applicability and significance in abstract algebraic representations.
 Keywords: Graph theory, padmakar-ivan index, randic index, non-coprime graphsMSC2020: 05C09

On p -Group Isomorphism: Search-to-Decision, Counting-to-Decision, and Nilpotency Class Reductions via Tensors

In this article, we study some classical complexity-theoretic questions regarding Group Isomorphism ( GpI ). We focus on p -groups (groups of prime power order) with odd p , which are believed to be a bottleneckcase for GpI , and work in the model of matrix groups over finite fields. Our main results are as follows: • Although search-to-decision and counting-to-decision reductions have been known for more than four decades for Graph Isomorphism , they had remained open for GpI , explicitly asked by Arvind and Torán ( EATCS Bull. , 2005). Extending methods from Tensor Isomorphism (TI) (Grochow and Qiao, ITCS 2021), we show moderately exponential-time such reductions within p -groups of class 2 and exponent p . • Despite the widely held belief that p -groups of class 2 and exponent p are the hardest cases of GpI , there was no reduction to these groups from any larger class of groups. Again using methods from TI (ibid.), we show the first such reduction, namely from isomorphismtesting of p -groups of “small” class and exponent p to those of class two and exponent p . For the first results, our main innovation is to develop linear-algebraic analogues of classical graph coloring gadgets, a key technique in studying the structural complexity of Graph Isomorphism . Unlike the graph coloring gadgets, which support restricting to various subgroups of the symmetric group, the problems we study require restricting to various subgroups of the general linear group, which entails significantly different and more complicated gadgets. The analysis of one of our gadgets relies on a classical result from group theory regarding random generation of classical groups (Kantor and Lubotzky, Geom. Dedicata , 1990). For the nilpotency class reduction, we combine a runtime analysis of the Lazard correspondence with TI-completeness results (Grochow and Qiao, ibid.).