Revisiting finite Abelian hidden subgroup problem and its distributed exact quantum algorithm

The study of automorphisms of algebraic structures plays a central role in understanding their internal symmetries and structural behavior. This work investigates the automorphism structure induced by <i>finite subgroups within infinite groups</i>, with particular emphasis on how these automorphisms can be characterized, classified, and effectively utilized. The focus is on the interaction between a finite subgroup and the ambient infinite group, analyzing how subgroup-preserving automorphisms extend to global automorphisms and how constraints imposed by finiteness influence the overall automorphism group. Special attention is given to classes of infinite groups such as abelian, conjugacies, and certain residually finite groups where finite subgroup automorphisms exhibit rich and tractable behavior. Building on this theoretical framework, this work explores <i>applications to symmetric cryptography,</i> where algebraic symmetry and complexity are essential for secure cryptographic design. Finite subgroup automorphisms are shown to provide a promising foundation for constructing cryptographic primitives, including key generation mechanisms, conjugacy-based encryption schemes, and secure mixing transformations. The inherent difficulty of reversing automorphism actions in large infinite groups, combined with the controlled structure of finite subgroups, offers a balance between computational efficiency and cryptographic strength. In overall, this work bridges abstract group theory and practical cryptographic applications, demonstrating that finite subgroup automorphisms of infinite groups constitute a viable and mathematically robust framework for advancing symmetric cryptographic systems.

We prove that the automorphism group of an affine, cubic surface with equation xyz=g(x,y) contains \mathbb{Z} as a finite index subgroup. These equations were first studied by Jacobsthal (1939) and Mordell (1952).

Unitary designs are essential tools in several quantum information protocols. Similarly to other design concepts, unitary designs are mainly used to facilitate averaging over a relevant space, in this case, the unitary group U ( d ) . While it is known that exact unitary t -designs exist for any degree t and dimension d , the most appealing type of designs, group designs (in which the elements of the design form a group), can provide at most 3 -designs. Moreover, even group 2 -designs can exist only in limited dimensions. In this paper, we present novel construction methods for creating exact generalized group designs based on the representation theory of the unitary group and its finite subgroups that overcome the 4 -design-barrier of unitary group designs. Furthermore, a construction is presented for creating generalized group 2 -designs in arbitrary dimensions.

We prove that the log canonical threshold of the base ideal of a complete linear system on a complex abelian variety is \ge 1 , and that equality holds if and only if the base locus has divisorial components. Consequently, the same assertions hold for the ideal of the intersection of translates of theta divisors by the points of a finite subgroup.

Abstract Color multiplets of the gauge fields and fermions on mass shell in SU (N ) Yang-Mills theory are classified according to representations of the Weyl group W (SU (N )). The multiplet structure of quark and gluon multiplets has been studied in the framework of a non-perturbative approach by considering complete exact equations of motion of the SU (N ) Yang-Mills theory with matter fields. An important case of singlet non-Abelian gluon solutions corresponding to one-dimensional singlet representations of the Weyl group is revised on a rigorous mathematical basis. We demonstrate that Weyl group as a finite color subgroup of SU (N ) reveals an inherent color symmetry of quarks and gluons on mass shell which determines a universal color multiplet structure of quark-gluon solutions in a pure SU (N ) Yang-Mills theory and in Abelian projected Yang-Mills theories with quarks. The obtained results allow to introduce strict concepts of fundamental particles, quarks and gluons, which differ drastically from the particle definitions in the conventional perturbative Yang-Mills theory. Possible applications of our results in non-perturbative quantum chromodynamics and hadron physics are discussed.

Abstract It is proved that, for an action of a discrete group having an element of infinite order on a path‐connected compact Hausdorff space, its equivariant asymptotic dimension with respect to the family of finite subgroups is at least two. Applying this result, we show that equivariant asymptotic dimension with respect to the family of finite subgroups can be strictly greater than dynamic asymptotic dimension.

Let \(\mathscr{G}\) be a finite multiplicative group of quaternion unit \(U(\mathbb{H})\). The \(\mathscr{G}\)-average of adjacency (resp., Laplacian) polynomial of a graph \(G\) is defined as the arithmetic mean of the characteristic polynomials of adjacency (resp., Laplacian) matrice of all \(\mathscr{G}\)-gain graphs on \(G\). In this paper, we prove that the \(\mathscr{G}\)-average adjacency (resp., Laplacian) polynomial of a graph for any non-trivial finite subgroup \(\mathscr{G}\) of \(U(\mathbb{H})\) coincides with its matching (resp., weighted TU-subgraph) polynomial, which generalizes previous findings for signed graphs.

Abstract We show that for a finite, nonempty subset A of a group, the quotient set A −1 A ≔ { a 1 −1 a 2 : a 1 , a 2 ∈ A } has size $$\mid A^{-1} A\mid \geq {5\over 3}\mid A \mid$$ ∣ A − 1 A ∣≥ 5 3 ∣ A ∣ , unless A is densely contained in a coset, or in a union of two cosets of a finite subgroup.

In this paper, we give a complete characterization of subgroup separability (also known as LERF property) for surface braid groups and virtual (singular) braid groups. More specifically, for a given surface S let [Formula: see text] be the n-string pure braid group of S. We prove that if S is large then [Formula: see text] is not LERF for [Formula: see text]. If S is not large, we prove [Formula: see text] is LERF if and only if S is not the Klein bottle; [Formula: see text] is LERF if and only if S is the disk, sphere or the projective plane; [Formula: see text] is LERF if and only if S is the sphere; and [Formula: see text] is not LERF for every [Formula: see text]. We note these results also hold for the full surface braid groups [Formula: see text], since [Formula: see text] is a finite index subgroup of [Formula: see text]. Finally, we prove that the virtual and virtual singular versions of those groups are LERF if and only if [Formula: see text].

Otto H. Kegel published two fundamental papers on Sylow Theory in locally finite groups: see [5] and [6]. The paper at hand summarises them, compares them and above all provides a list of their open issues which are still open until the present day. To study crucial configurations, Kegel developed in [6] the quite excogitated concept of the “(smooth simple straight) split sequences of finite p-perfect subgroups with their associated ascending sequences of subgroups ” which is related to his equally very fine concept of the “Sylow-separated (ascending) sequences of p-subgroups with associated sequences of Sylow p-subgroups ” he had developed already more then ten years earlier in [5]. When I met him personally in July 2022, I tried to find out how he detected these really ingenious ideas, but woefully was unsuccessful, although I was really very privileged to witness up very close the creation of the idea of [6] (see [3]). A scheduled new attempt in July 2025 (see [3]) could not be realised …. Mathematics Subject Classification (2020): 20D20, 20F50, 20D15, 20D06, 20D10

Abstract Let V be a general three-dimensional quartic in the complex projective space ℙ 4 such that the only singularity of V is a double point of rank 3. We prove that V is a birationally rigid variety. Its group of birational self-maps is, up to the finite subgroup of biregular automorphisms, a free product of 25 cyclic groups of order 2. It follows that the complement to the set of birationally rigid factorial quartics with terminal singularities is of codimension at least 3 in the natural parameter space.

We classify two-dimensional right-angled Coxeter groups that are quasiisometric to a right-angled Artin group defined by a tree, and show that when this is true the right-angled Coxeter group actually contains a visible finite index right-angled Artin subgroup.

Abstract For Γ a finite subgroup of $\mathrm{SL}_2(\mathbb{C})$ and $n \geq 1$ , we study the fibres of the Procesi bundle over the Γ-fixed points of the Hilbert scheme of n points in the plane. For each irreducible component of this fixed point locus, our approach reduces the study of the fibres of the Procesi bundle, as an $(\mathfrak{S}_n \times \Gamma)$ -module, to the study of the fibres of the Procesi bundle over an irreducible component of dimension zero in a smaller Hilbert scheme. When Γ is of type A , our main result shows, as a corollary, that the fibre of the Procesi bundle over the monomial ideal associated with a partition λ is induced, as an $(\mathfrak{S}_n \times \Gamma)$ -module, from the fibre of the Procesi bundle over the monomial ideal associated with the core of λ . We give different proofs of this corollary in two edge cases using only representation theory and symmetric functions.

Let E be an elliptic curve over a number field L and for a finite set S of primes, let ρ E,S :Gal(L ¯/L)→GL 2 (ℤ S ) be the S-adic Galois representation. If L∩ℚ(ζ n )=ℚ for all positive integers n whose prime factors are in S, then detρ E,S :Gal(L ¯/L)→ℤ S × is surjective. We say that a finite index subgroup H⊆GL 2 (ℤ S ) is minimal if det:H→ℤ S × is surjective, but det:K→ℤ S × is not surjective for any proper closed subgroup K of H. We show that there are no minimal subgroups of GL 2 (ℤ S ) unless S={2}, while minimal subgroups of GL 2 (ℤ 2 ) are plentiful. We give models for all the genus 0 modular curves associated to minimal subgroups of GL 2 (ℤ 2 ), and construct an infinite family of elliptic curves over imaginary quadratic fields with bad reduction only at 2 and with minimal 2-adic image.

We study relatively minimal surfaces equipped with a strongly isotrivial elliptic fibration in positive characteristic by means of the notion of equivariantly normal curves introduced and developed recently by Brion in [Pure Appl. Math. Q. 20 (2024), 1065–1095 and arXiv:2405.12020v1]. Such surfaces are isomorphic to a contracted product E\times^{G} X , where E is an elliptic curve, G is a finite subgroup scheme of E and X is a G -normal curve. Using this description, we compute their Betti numbers to determine their birational classes. This allows us to complete the classification of maximal automorphism groups of surfaces in any characteristic, extending the result in characteristic zero obtained in [Ann. Inst. Fourier (Grenoble) 74 (2024), 545–587]. When G is diagonalizable, we compute additional invariants to study the structure of their Picard schemes.

Abstract Let $G\stackrel{\alpha }{\curvearrowright }(M,\tau )$ be a trace-preserving action of a finite group $G$ on a tracial von Neumann algebra. Suppose that $A \subset M$ is a finitely generated $\alpha $-invariant unital *-subalgebra. We give a formula relating the von Neumann dimension of the space of derivations on $A$ valued on its coarse bimodule to the von Neumann dimension of the space of derivations on $A \rtimes ^{\textrm{alg}}_\alpha G$ valued on its coarse bimodule, which is reminiscent of Schreier’s formula for finite index subgroups of free groups. This formula induces a formula for the free Stein dimension (defined by Charlesworth and Nelson), $\dim \textsf{Der}_{c,1\otimes 1}(A,\tau )$, and $\Delta $ (defined by Connes and Shlyakhtenko). The latter is done by establishing that $\Delta $ is equal to the von Neumann dimension of a certain subspace of the derivation space of $A$, similar to that of the free Stein dimension. Using the formula for $\Delta $, we strengthen the recent results of Shlyakhtenko on the microstates free entropy dimension.

Finite non-parabolic subgroups of relatively hyperbolic groups

We show that the probability for a finitely generated subgroup of the modular group, of size n , to be almost malnormal or non-parabolic, tends to 0 as n tends to infinity – where the notion of the size of a subgroup is based on a natural graph-theoretic representation of the subgroup. The proofs of these results rely on the combinatorial and asymptotic study of a natural map, which associates with any finitely generated subgroup of \mathsf{PSL}_{2}(\mathbb{Z}) a graph which we call its silhouette, which can be interpreted as a conjugacy class of free finite index subgroups of \mathsf{PSL}_{2}(\mathbb{Z}) .

For any right-angled Artin group A , Charney, Stambaugh and Vogtmann showed that the subgroup U 0 .A / Out.A / generated by Whitehead automorphisms and inversions acts properly and cocompactly on a contractible space K .We show that any finite subgroup of U 0 .A / fixes a point of K .This generalizes the fact that any finite subgroup of Out.F n / fixes a point of outer space, and implies that there are only finitely many conjugacy classes of finite subgroups in U 0 .A /. 20F28, 20F36, 20F65