Galois covers of ℙ1 and number fields with large class groups

For each finite subgroup [Formula: see text] of [Formula: see text], and for each integer [Formula: see text] coprime to [Formula: see text], we construct explicitly infinitely many Galois extensions of [Formula: see text] with group [Formula: see text] and whose ideal class group has [Formula: see text]-rank at least [Formula: see text]. This gives new [Formula: see text]-rank records for class groups of number fields.

Clifford Circuit Optimization with Templates and Symbolic Pauli Gates

The Clifford group is a finite subgroup of the unitary group generated by the Hadamard, the CNOT, and the Phase gates. This group plays a prominent role in quantum error correction, randomized benchmarking protocols, and the study of entanglement. Here we consider the problem of finding a short quantum circuit implementing a given Clifford group element. Our methods aim to minimize the entangling gate count assuming all-to-all qubit connectivity. First, we consider circuit optimization based on template matching and design Clifford-specific templates that leverage the ability to factor out Pauli and SWAP gates. Second, we introduce a symbolic peephole optimization method. It works by projecting the full circuit onto a small subset of qubits and optimally recompiling the projected subcircuit via dynamic programming. CNOT gates coupling the chosen subset of qubits with the remaining qubits are expressed using symbolic Pauli gates. Software implementation of these methods finds circuits that are only 0.2% away from optimal for 6 qubits and reduces the two-qubit gate count in circuits with up to 64 qubits by 64.7% on average, compared with the Aaronson-Gottesman canonical form.

On Periodic Groups Saturated with Finite Frobenius Groups

A group is called weakly conjugate biprimitively finite if each its element of prime order generates a finite subgroup with any of its conjugate elements. A binary finite group is a periodic group in which any two elements generate a finite subgroup. If $\mathfrak{X}$ is some set of finite groups, then the group $G$ saturated with groups from the set $\mathfrak{X}$ if any finite subgroup of $G$ is contained in a subgroup of $G$, isomorphic to some group from $\mathfrak{X}$. A group $G = F \leftthreetimes H$ is a Frobenius group with kernel $F$ and a complement $H$ if $H \cap H^f = 1$ for all $f \in F^{\#}$ and each element from $G \setminus F$ belongs to a one conjugated to $H$ subgroup of $G$. In the paper we prove that a saturated with finite Frobenius groups periodic weakly conjugate biprimitive finite group with a nontrivial locally finite radical is a Frobenius group. A number of properties of such groups and their quotient groups by a locally finite radical are found. A similar result was obtained for binary finite groups with the indicated conditions. Examples of periodic non locally finite groups with the properties above are given, and a number of questions on combinatorial group theory are raised.

On Two Properties of Shunkov Group

One of the interesting classes of mixed groups ( i.e. groups that can contain both elements of finite order and elements of infinite order) is the class of Shunkov groups. The group $G$ is called Shunkov group if for any finite subgroup $H$ of $G$ in the quotient group $N_G(H)/H$, any two conjugate elements of prime order generate a finite group. When studying the Shunkov group $G$, a situation often arises when it is necessary to move to the quotient group of the group $G$ by some of its normal subgroup $N$. In which cases is the resulting quotient group $G/N$ again a Shunkov group? The paper gives a positive answer to this question, provided that the normal subgroup $N$ is locally finite and the orders of elements of the subgroup $N$ are mutually simple with the orders of elements of the quotient group $G/N$. Let $ \mathfrak{X}$ be a set of groups. A group $G$ is saturated with groups from the set $ \mathfrak{X}$ if any finite subgroup of $G$ is contained in a subgroup of $ G$ that is isomorphic to some group of $\mathfrak{X}$ . If all elements of finite orders from the group $G$ are contained in a periodic subgroup of the group $G$, then it is called the periodic part of the group $G$ and is denoted by $T(G)$. It is proved that the Shunkov group saturated with finite linear and unitary groups of degree 3 over finite fields has a periodic part that is isomorphic to either a linear or unitary group of degree 3 on a suitable locally finite field. \end{abstracte}

On Periodic Shunkov’s Groups with Almost Layer-finite Normalizers of Finite Subgroups

Layer-finite groups first appeared in the work by S. N. Chernikov (1945). Almost layer-finite groups are extensions of layer-finite groups by finite groups. The class of almost layer-finite groups is wider than the class of layer-finite groups; it includes all Chernikov groups, while it is easy to give examples of Chernikov groups that are not layer-finite. The author develops the direction of characterizing well-known and well-studied classes of groups in other classes of groups with some additional (rather weak) finiteness conditions. A Shunkov group is a group 𝐺 in which for any of its finite subgroups 𝐾 in the quotient group <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msub><mi>N</mi><mi>G</mi></msub><mo>(</mo><mi>K</mi><mo>)</mo></mrow><mi>K</mi></mfrac></math> any two conjugate elements of prime order generate a finite subgroup. In this paper, we prove the properties of periodic not almost layer-finite Shunkov groups with condition: the normalizer of any finite nontrivial subgroup is almost layer-finite. Earlier, these properties were proved in various articles of the author, as necessary, sometimes under some conditions, then under others (the minimality conditions for not almost layer-finite subgroups, the absence of second-order elements in the group, the presence of subgroups with certain properties in the group). At the same time, it was necessary to make remarks that this property is proved in almost the same way as in the previous work, but under different conditions. This eliminates the shortcomings in the proofs of many articles by the author, in which these properties are used without proof.

Instantons and Bows for the Classical Groups

The construction of Atiyah, Drinfeld, Hitchin and Manin provided complete description of all instantons on Euclidean four-space. It was extended by Kronheimer and Nakajima to instantons on ALE spaces, resolutions of orbifolds $\mathbb{R}^4/\Gamma$ by a finite subgroup Γ⊂SU(2). We consider a similar classification, in the holomorphic context, of instantons on some of the next spaces in the hierarchy, the ALF multi-Taub-NUT manifolds, showing how they tie in to the bow solutions to Nahm’s equations via the Nahm correspondence. Recently Nakajima and Takayama constructed the Coulomb branch of the moduli space of vacua of a quiver gauge theory, tying them to the same space of bow solutions. One can view our construction as describing the same manifold as the Higgs branch of the mirror gauge theory as described by Cherkis, O’Hara and Saemann. Our construction also yields the monad construction of holomorphic instanton bundles on the multi-Taub-NUT space for any classical compact Lie structure group.

ON THE DIMENSION OF GROUPS THAT SATISFY CERTAIN CONDITIONS ON THEIR FINITE SUBGROUPS

We say a group G satisfies properties (M) and (NM) if every nontrivial finite subgroup of G is contained in a unique maximal finite subgroup, and every nontrivial finite maximal subgroup is self-normalizing. We prove that the Bredon cohomological dimension and the virtual cohomological dimension coincide for groups that admit a cocompact model for EG and satisfy properties (M) and (NM). Among the examples of groups satisfying these hypothesis are cocompact and arithmetic Fuchsian groups, one-relator groups, the Hilbert modular group, and 3-manifold groups.

BOUNDEDNESS PROPERTIES OF AUTOMORPHISM GROUPS OF FORMS OF FLAG VARIETIES

We call a flag variety admissible if its automorphism group is the projective general linear group. (This holds in most cases.) Let K be a field of characteristic 0, containing all roots of unity. Let the K-variety X be a form of an admissible flag variety. We prove that X is either ruled, or the automorphism group of X is bounded, meaning that there exists a constant C ∈ ℕ such that if G is a finite subgroup of AutK(X), then the cardinality of G is smaller than C.