Classification of four qubit states and their stabilisers under SLOCC operations

We classify four qubit states under SLOCC operations, that is, we classify the orbits of the group SL(2,C)^4 on the Hilbert space H_4 = (C^2)^{\otimes 4}. We approach the classification by realising this representation as a symmetric space of maximal rank. We first describe general methods for classifying the orbits of such a space. We then apply these methods to obtain the orbits in our special case, resulting in a complete and irredundant classification of SL(2,C)^4-orbits on H_4. It follows that an element of H_4 is conjugate to an element of precisely 87 classes of elements. Each of these classes either consists of one element or of a parametrised family of elements, and the elements in the same class all have equal stabiliser in SL(2,C)^4. We also present a complete and irredundant classification of elements and stabilisers up to the action of the semidirect product Sym_4\ltimes\SL(2,C)^4 where Sym_4 permutes the four tensor factors of H_4.

Finite 𝒟C-groups

Let [Formula: see text] be a group and [Formula: see text]. [Formula: see text] is said to be a [Formula: see text]-group if [Formula: see text] is a chain under set inclusion. In this paper, we prove that a finite [Formula: see text]-group is a semidirect product of a Sylow [Formula: see text]-subgroup and an abelian [Formula: see text]-subgroup. For the case of [Formula: see text] being a finite [Formula: see text]-group, we obtain an optimal upper bound of [Formula: see text] for a [Formula: see text] [Formula: see text]-group [Formula: see text]. We also prove that a [Formula: see text] [Formula: see text]-group is metabelian when [Formula: see text] and provide an example showing that a non-abelian [Formula: see text] [Formula: see text]-group is not necessarily metabelian when [Formula: see text]. In particular, [Formula: see text] [Formula: see text]-groups are characterized.

MAKE: A matrix action key exchange

We offer a public key exchange protocol based on a semidirect product of two cyclic (semi)groups of matrices over Z p {{\mathbb{Z}}}_{p} . One of the (semi)groups is additive, and the other one is multiplicative. This allows us to take advantage of both operations on matrices to diffuse information. We note that in our protocol, no power of any matrix or of any element of Z p {{\mathbb{Z}}}_{p} is ever exposed, so standard classical attacks on Diffie–Hellman-like protocols are not applicable.

Group-theoretical graph categories

The semidirect product of a finitely generated group dual with the symmetric group can be described through so-called group-theoretical categories of partitions (covers only a special case; due to Raum–Weber, 2015) and skew categories of partitions (more general; due to Maaßen, 2018). We generalize these results to the case of graph categories, which allows to replace the symmetric group by the group of automorphisms of some graph.

Intense Automorphisms of Finite Groups

Let G G be a group. An automorphism of G G is called intense if it sends each subgroup of G G to a conjugate; the collection of such automorphisms is denoted by Int ⁡ ( G ) \operatorname {Int}(G) . In the special case in which p p is a prime number and G G is a finite p p -group, one can show that Int ⁡ ( G ) \operatorname {Int}(G) is the semidirect product of a normal p p -Sylow and a cyclic subgroup of order dividing p − 1 p-1 . In this paper we classify the finite p p -groups whose groups of intense automorphisms are not themselves p p -groups. It emerges from our investigation that the structure of such groups is almost completely determined by their nilpotency class: for p > 3 p>3 , they share a quotient, growing with their class, with a uniquely determined infinite 2-generated pro- p p group.

Quasi-decompositions and quasidirect products of Hilbert algebras

A quasi-decomposition of a Hilbert algebra A is a pair (C, D) of its subalgebras such that (i) every element a ∈ A is a meet c ∧ d with c ∈ C, d ∈ D, where c and d are compatible (i.e., c → d = c → (c ∧ d)), and (ii) d → c = c (then c is uniquely defined). Quasi-decompositions are intimately related to the so-called triple construction of Hilbert algebras, which we reinterpret as a construction of quasidirect products. We show that it can be viewed as a generalization of the semidirect product construction, that quasidirect products has a certain universal property and that they can be characterised in terms of short exact sequences. We also discuss four classes of Hilbert algebras and give for each of them conditions on a quasi-decomposition of an arbitrary Hilbert algebra A under which A belongs to this class.

Chief factors in Polish groups

In finite group theory, chief factors play an important and well-understood role in the structure theory. We here develop a theory of chief factors for Polish groups. In the development of this theory, we prove a version of the Schreier refinement theorem. We also prove a trichotomy for the structure of topologically characteristically simple Polish groups. The development of the theory of chief factors requires two independently interesting lines of study. First we consider injective, continuous homomorphisms with dense normal image. We show such maps admit a canonical factorisation via a semidirect product, and as a consequence, these maps preserve topological simplicity up to abelian error. We then define two generalisations of direct products and use these to isolate a notion of semisimplicity for Polish groups.

On the group of unit-valued polynomial functions

Let R be a finite commutative ring. The set $${{\mathcal{F}}}(R)$$ F ( R ) of polynomial functions on R is a finite commutative ring with pointwise operations. Its group of units $${{\mathcal{F}}}(R)^\times $$ F ( R ) × is just the set of all unit-valued polynomial functions. We investigate polynomial permutations on $$R[x]/(x^2)=R[\alpha ]$$ R [ x ] / ( x 2 ) = R [ α ] , the ring of dual numbers over R, and show that the group $${\mathcal{P}}_{R}(R[\alpha ])$$ P R ( R [ α ] ) , consisting of those polynomial permutations of $$R[\alpha ]$$ R [ α ] represented by polynomials in R[x], is embedded in a semidirect product of $${{\mathcal{F}}}(R)^\times $$ F ( R ) × by the group $${\mathcal{P}}(R)$$ P ( R ) of polynomial permutations on R. In particular, when $$R={\mathbb{F}}_q$$ R = F q , we prove that $${\mathcal{P}}_{{\mathbb{F}}_q}({\mathbb{F}}_q[\alpha ])\cong {\mathcal{P}}({\mathbb{F}}_q) \ltimes _\theta {{\mathcal{F}}}({\mathbb{F}}_q)^\times $$ P F q ( F q [ α ] ) ≅ P ( F q ) ⋉ θ F ( F q ) × . Furthermore, we count unit-valued polynomial functions on the ring of integers modulo $${p^n}$$ p n and obtain canonical representations for these functions.