Abstract We give a new method for counting extensions of a number field asymptotically by discriminant, which we employ to prove many new cases of Malle’s Conjecture and counterexamples to Malle’s Conjecture. We consider families of extensions whose Galois closure is a fixed permutation group 𝐺. Our method relies on having asymptotic counts for 𝑇-extensions for some normal subgroup 𝑇 of 𝐺, uniform bounds for the number of such 𝑇-extensions, and possibly weak bounds on the asymptotic number of G / T G/T -extensions. However, we do not require that most 𝑇-extensions of a G / T G/T -extension are 𝐺-extensions. Our new results use 𝑇 either abelian or S 3 m S_{3}^{m} , though our framework is general.

Vertex-transitive graphs with small motion and transitive permutation groups with small minimal degree

Abstract We prove that if is a transitive permutation group of sufficiently large degree , then either is primitive and Frobenius, or the proportion of derangements in is larger than . This is sharp, generalizes substantially bounds of Cameron–Cohen and Guralnick–Wan, and settles conjectures of Guralnick–Tiep and Bailey–Cameron–Giudici–Royle in large degree. We also give an application to coverings of varieties over finite fields.

Groups of permutations that are even on subsets of a fixed size, and related monoids

For [Formula: see text] with [Formula: see text], let [Formula: see text] denote the maximum number of edge disjoint trees connecting [Formula: see text] in [Formula: see text]. For [Formula: see text], the generalized [Formula: see text]-connectivity [Formula: see text] of an [Formula: see text]-vertex connected graph [Formula: see text] is defined to be [Formula: see text]. The generalized [Formula: see text]-connectivity can serve for measuring the fault tolerance of an interconnection network. The bubble-sort graph [Formula: see text] for [Formula: see text] is a Cayley graph over the symmetric group of permutations on [Formula: see text] generated by transpositions from the set [Formula: see text]. In this paper, we show that for the bubble-sort graphs [Formula: see text] with [Formula: see text], [Formula: see text].

Let ( G , X ) be a non-geometric sharp permutation group of type { 0 , k } and let x ∈ X . We analyze the structure of ( G , X ) under the assumption that the point stabilizer G x ≅ K : L is a Frobenius group with elementary abelian Frobenius kernel K ≅ C p × C p , p an odd prime, and Frobenius complement L where L acts reducibly on K. In particular, for each odd prime p we determine all viable values of k and |X|, and for each of these, we determine the orbit structure of G x . We also completely describe the structure of ( G , X ) when k is as large as possible.

New simple solutions of the Yang–Baxter equation and their permutation groups

Abstract We construct a new local Poisson bracket compatible with the second unconstrained Adler-Gelfand-Dickey bracket. The resulting bihamiltonian structure admits a dispersionless limit and the leading term defines a logarithmic Dubrovin–Frobenius manifold. Furthermore, we show that this Dubrovin-Frobenius manifold can be constructed on the orbits space of the standard representation of the permutation group.

Abstract A transitive permutation group is called semiprimitive if each of its normal subgroups is either semiregular or transitive. The class of semiprimitive groups properly includes primitive groups, quasiprimitive groups and innately transitive groups. The latter three classes of rank 3 permutation groups have been classified, making significant progress towards solving the long-standing problem of classifying permutation groups of rank 3. In this paper, we complete the classification of finite semiprimitive groups of rank 3, building on the recent work of Huang, Li and Zhu. Examples include Schur coverings of certain almost simple 2-transitive groups and three exceptional small groups.

N-ary groups of panmagic permutations from the Post coset theorem

Corrigendum and addendum to “Transitive permutation groups where nontrivial elements have at most two fixed points” [J. Pure Appl. Algebra 219(4) (2015) 729–759

Abstract Symmetries are omnipresent in physics and have been used to reduce the number of degrees of freedom of systems. In this work, we investigate the properties of M SU ( n ) , M -invariant subspaces of the special unitary Lie group SU(n). This group is relevant to quantum computing and quantum systems in general. We demonstrate that for certain choices of M , the subset M SU ( n ) inherits many topological and group properties from SU(n). We then present a combinatorial method for computing the dimension of such subspaces when M is a representation of a permutation group acting on a tensor product of spaces G SU ( n ) , or a Hamiltonian H ( n ) SU ( n ) . The Kronecker product of su ( 2 ) matrices is employed to construct the Lie algebras associated with different permutation-invariant groups GSU(n). Numerical results on the number of dimensions support the developed theory.

The 15-puzzle is a classic sliding puzzle consisting of a 4x4 grid with 15 numbered square tiles and one empty space. In 1974, Wilson generalized the 15-puzzle to find the group of permutations on graphs. In this work, we provide a variation of a proof of Wilsons theorem, propose a result for 1-connected and disconnected graphs, find a new manual algorithm for solving sliding graph puzzles, and extend existing computer algorithms on the 15-puzzle to solve any sliding graph puzzle.

The newly developed automorphism ensemble decoder (AED) leverages the rich automorphisms of Reed–Muller (RM) codes to achieve near maximum likelihood (ML) performance at short code lengths. However, the performance gain of AED comes at the cost of high complexity, as the ensemble size required for near ML decoding grows exponentially with the code length. In this work, we address this complexity issue by focusing on the factor graph permutation group (FGPG), a subgroup of the full automorphism group of RM codes, to generate permutations for AED. We propose a uniform partitioning of FGPG based on the affine bijection permutation matrices of automorphisms, where each subgroup of FGPG exhibits permutation invariance (PI) in a Plotkin construction-based information set partitioning for RM codes. Furthermore, from the perspective of polar codes, we exploit the PI property to prove a subcode estimate convergence (SEC) phenomenon in the AED that utilizes successive cancellation (SC) or SC list (SCL) constituent decoders. Observing that strong SEC correlates with low noise levels, where the full decoding capacity of AED is often unnecessary, we perform path pruning to reduce the decoding complexity without compromising the performance. Our proposed SEC-aided path pruning allows only a subset of constituent decoders to continue decoding when the intensity of SEC exceeds a preset threshold during decoding. Numerical results demonstrate that, for the FGPG-based AED of various short RM codes, the proposed SEC-aided path pruning technique incurs negligible performance degradation, while achieving a complexity reduction of up to 67.6%.

Abstract We prove the vanishing of bounded cohomology of the groups acting on trees with almost prescribed local actions , where are finite permutation groups such that is 2‐transitive. In contrast, when is not 2‐transitive, we prove that the second bounded cohomology with real coefficients of the groups is infinite‐dimensional.

Herein, a matrix representation of the Hamilton quaternion group by 4 × 4 square matrices with entries equal to −1, 0, or 1 is defined. It is proven that this group, denoted as QM,∘, is a group of rotational symmetries of the four-dimensional hypercube ℤ24, that is, a subgroup of the special orthogonal group SO4ℤ. As a consequence, QM,∘ is a group of rotational symmetries for each of the biological hypercubes RNY, YNY, YNR, and RNR. It is also proven that QM,∘ is a group of permutations of the eight cubes contained in the four-dimensional hypercube ℤ24. The latter is a novel result. It is also proven that the matrix quaternion group QM,∘ is a normal subgroup of SO4ℤ and that the latter is a semidirect product of QM,∘ with a copy of the special orthogonal group SO3ℤ, also called an octahedral group because it is a group of rotational symmetries of a regular octahedron or of a three-dimensional cube.

Nonreciprocity and permutation symmetry, despite their mutual exclusivity in two-component systems, coexist in multicomponent biological and synthetic systems, ranging from neural circuits to active metamaterials. While these phenomena have been extensively studied independently in the contexts of nonreciprocal phase transitions (e.g., flocking, synchronization, and pattern formation) and cluster synchronization (e.g., oscillator networks), their synergistic effects remain largely unexplored. This study introduces a framework, integrating mean-field theory and group representation theory, to investigate collective behaviors in synchronization and pattern formation systems exhibiting both nonreciprocity and permutation symmetry. Analyzing two three-species scenarios with S_{2} (transposition) and C_{3} (cyclic) permutation groups, we identify distinct symmetry-constrained traveling wave phases in synchronization system. Both phases exhibit parity-time symmetry breaking, but their dynamics and phase transitions diverge due to differing symmetry constraints. Spontaneous permutation symmetry breaking transforms the phase transition points of the traveling wave phase from exceptional points to Hopf bifurcations. Furthermore, we discover a time-dependent phase in scenario with S_{2}, absent in two-species systems. This phase features a dynamic interplay between unidirectional, time-crystal-like oscillations in one species and symmetrical pendulumlike motions in the other two. Our results demonstrate that permutation symmetry significantly expands the repertoire of symmetry-constrained collective behaviors and phase transitions in nonreciprocal systems, exceeding the constraints of two-species systems. The identified mechanism, symmetry-constrained organization of nonreciprocal fluxes, provides a design principle for engineering collective behaviors in neuronal circuits and active metamaterials.

Viennot shadows and graded module structure in colored permutation groups

A Young subgroup of the symmetric group $\mathcal{S}_{N}$, the permutation group of $\{ 1,2,\dots,N\} $, is generated by a subset of the adjacenttranspositions $\{ ( i,i+1) \mid 1\leq i < N\}$. Such a group is realized as the stabilizer $G_{n}$ of a monomial $x^{\lambda}$ $\big({=}\,x_{1}^{\lambda_{1}}x_{2}^{\lambda_{2}}\cdots x_{N}^{\lambda_{N}}\big)$ with ${\lambda=\bigl( d_{1}^{n_{1}},d_{2}^{n_{2}}, \dots,d_{p}^{n_{p}}\bigr)} $ (meaning $d_{j}$ is repeated $n_{j}$ times, $1\leq j\leq p$, and $d_{1}>d_{2}>\dots>d_{p}\geq0$), thus is isomorphic to the direct product $\mathcal{S}_{n_{1}}\times\mathcal{S}_{n_{2}} \times\cdots\times\mathcal{S}_{n_{p}}$. The interval $\{ 1,2,\dots,N\} $ is a union of disjoint sets $I_{j}= \{ i\mid \lambda_{i}=d_{j} \} $. The orbit of $x^{\lambda}$ under the action of $\mathcal{S}_{N}$ (by permutation of coordinates) spans a module $V_{\lambda}$, the representation induced from the identity representation of $G_{n}$. The space $V_{\lambda}$ decomposes into a direct sum of irreducible $\mathcal{S}_{N}$-modules. The spherical function is defined for each of these, it is the character of the module averaged over the group $G_{n}$. This paper concerns the value of certain spherical functions evaluated at a cycle which has no more than one entry in each interval $I_{j}$. These values appear in the study of eigenvalues of the Heckman-Polychronakos operators in the paper by V. Gorin and the author [arXiv:2412:01938]. In particular, the present paper determines the spherical function value for $\mathcal{S}_{N}$-modules of hook tableau type, corresponding to Young tableaux of shape $\bigl[ N-b,1^{b}\bigr]$.

The two basic puzzles in Rubik’s cube can be summarized as recognizing solvable configurations and finding a common solution for valid configurations. Indeed, in the process of solving the first puzzle, we are required to come up with a common solution. The mathematical structure of moves in Rubik’s cube is also famous. This paper clarifies the difference between process and essence through an original concept: effect, and redefines the operation group G in a strict way. It also corrects a widespread mistake in pinpointing the mapping type between the operation group G and permutation groups, which appears in some former papers. The effect system grasps the essence of operations, introduces reference frames in physics through pure mathematics, and specifically facilitates the solution for the basic puzzles in Rubik’s cube and megaminx. The effect system has the potential to be utilized in physically symmetric structures with moves that permute mathematically analyzable configurations.