Automorphisms of n-Ary Groups

Isotopies and autotopies of n-ary groups are described. As a consequence, we obtain various characterizations of the group of automorphisms of n-ary groups. We also determine the number of automorphisms of a given n-ary group.

Distance-transitive strongly regular graphs

We present a complete description of strongly regular graphs admitting a distance-transitive group of automorphisms. Parts of the list have already appeared in the literature; however, this is the first time that the complete list appears in one place. The description is complemented, where possible, with the discussion of the corresponding distance-transitive groups and some further properties of the graphs. We also point out an open problem.

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.

An infinite-dimensional 2-generated primitive axial algebra of Monster type

Rehren proved in Axial algebras. Ph.D. thesis, University of Birmingham (2015), Trans Am Math Soc 369:6953–6986 (2017) that a primitive 2-generated axial algebra of Monster type $$(\alpha ,\beta )$$ ( α , β ) , over a field of characteristic other than 2, has dimension at most 8 if $$\alpha \notin \{2\beta ,4\beta \}$$ α ∉ { 2 β , 4 β } . In this note, we show that Rehren’s bound does not hold in the case $$\alpha =4\beta $$ α = 4 β by providing an example (essentially the unique one) of an infinite-dimensional 2-generated primitive axial algebra of Monster type $$(2,\frac{1}{2})$$ ( 2 , 1 2 ) over an arbitrary field $${{\mathbb {F}}}$$ F of characteristic other than 2 and 3. We further determine its group of automorphisms and describe some of its relevant features.

the complexification of the exceptional Jordan algebra and applications to particle physics

Recent papers of Todorov and Dubois-Violette[4] and Krasnov[7] contributed revitalizing the study of the exceptional Jordan algebra h3(O) in its relations with the true Standard Model gauge group GSM. The absence of complex representations of F4 does not allow Aut (h3 (O)) to be a candidate for any Grand Unified Theories, but the group of automorphisms of the complexification of this algebra isisomorphic to the compact form of E6. Following Boyle in [12], it is then easy to show that the gauge group of the minimal left-right symmetric extension of the Standard Model is isomorphic to a proper subgroup of Aut(C⊗h3(O))

On the solvability of graded Novikov algebras

We show that the right ideal of a Novikov algebra generated by the square of a right nilpotent subalgebra is nilpotent. We also prove that a [Formula: see text]-graded Novikov algebra [Formula: see text] over a field [Formula: see text] with solvable [Formula: see text]-component [Formula: see text] is solvable, where [Formula: see text] is a finite additive abelean group and the characteristic of [Formula: see text] does not divide the order of the group [Formula: see text]. We also show that any Novikov algebra [Formula: see text] with a finite solvable group of automorphisms [Formula: see text] is solvable if the algebra of invariants [Formula: see text] is solvable.

Symmetric 1-designs from PGL2(q), for q an odd prime power

All non-trivial point and block-primitive 1-(v, k, k) designs 𝓓 that admit the group G = PGL2(q), where q is a power of an odd prime, as a permutation group of automorphisms are determined. These self-dual and symmetric 1-designs are constructed by defining { |M|/|M ∩ Mg|: g ∈ G } to be the set of orbit lengths of the primitive action of G on the conjugates of M.

Error correction by Reed–Solomon codes using its automorphisms

The article explores the syndrome invariants of АГ-group of automorphisms of Reed–Solomon codes (RS-codes) that are a joint group of affine and cyclic permutations. The found real invariants are a set of norms of N Г-orbits that make up one or another АГ-orbit. The norms of Г-orbits are vectors with 2 1 Cδ− coordinates from the Galois field, that are determined by all kinds of pairs of components of the error syndromes. In this form, the invariants of the АГ-orbits were cumbersome and difficult to use. Therefore, their replacement by conditional partial invariants is proposed. These quasi-invariants are called norm-projections. Norm-projection uniquely identifies its АГ-orbit and therefore serves as an adequate way for formulating the error correction method by RS-codes based on АГ-orbits. The power of the АГ-orbits is estimated by the value of N2, equal to the square of the length of the RS-code. The search for error vectors in transmitted messages by a new method is reduced to parsing the АГ‑orbits, but actually their norm-projections, with the subsequent search for these errors within a particular АГ-orbit. Therefore, the proposed method works almost N2 times faster than traditional syndrome methods, operating on the basic of the “syndrome – error” principle, that boils down to parsing the entire set of error vectors until a specific vector is found.

The upper central series of the maximal normal 𝑝-subgroup of a group of automorphisms

Let 𝐺 be a bounded abelian 𝑝-group, with automorphism group Aut ⁡ ( G ) \operatorname{Aut}(G) . Whenever 𝐺 satisfies certain conditions, we determine the upper central series and nilpotency class of the maximal normal 𝑝-subgroup of Aut ⁡ ( G ) \operatorname{Aut}(G) .

High rank torus actions on contact manifolds

We prove LeBrun–Salamon conjecture in the following situation: if X is a contact Fano manifold of dimension $$2n+1$$ 2 n + 1 whose group of automorphisms is reductive of rank $$\ge \max (2,(n-3)/2)$$ ≥ max ( 2 , ( n - 3 ) / 2 ) then X is the adjoint variety of a simple group. The rank assumption is fulfilled not only by the three series of classical linear groups but also by almost all the exceptional ones.