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Axial algebras are a class of commutative non-associative algebras generated by idempotents, called axes, with adjoint action semi-simple and satisfying a prescribed fusion law. Axial algebras were introduced by Hall, Rehren and Shpectorov (in 2015) as a broad generalization of Majorana algebras of Ivanov, whose axioms were derived from the properties of the Griess algebra for the Monster sporadic simple group. The class of axial algebras of Monster type includes Majorana algebras for the Monster and many other sporadic simple groups, Jordan algebras for classical and some exceptional simple groups, and Matsuo algebras corresponding to 3-transposition groups. Thus, axial algebras of Monster type unify several strands in the theory of finite simple groups. It is shown here that double axes, i.e., sums of two orthogonal axes in a Matsuo algebra, satisfy the fusion law of Monster type (2 η , η). Primitive subalgebras generated by two single or double axes are completely classified and 3-generated primitive subalgebras are classified in one of the three cases. These classifications further lead to the general flip construction outputting a rich variety of axial algebras of Monster type. An application of the flip construction to the case of Matsuo algebras related to the symmetric groups results in three new explicit infinite series of such algebras. [ABSTRACT FROM AUTHOR]

Hurwitz spaces are moduli spaces of curve covers. The isomorphism classes of covers of P¹ℂ with given ramification data are parameterized combinatorially by Nielsen tuples in the monodromy group G. The Artin braid group acts on Nielsen tuples, and the orbits of this action correspond to the connected components of the corresponding Hurwitz space. In this article we consider the case G = A5. We give a complete classification of the braid orbits for all ramification types, showing that the components are always distinguishable by the Fried-Serre lift invariant. [ABSTRACT FROM AUTHOR]

In Pralle and Shpectorov (Adv Geom 7(1):1-17, ) the class of ovoidal hyperplanes in dual polar spaces of rank 4 is described. In this paper we observe that by removing such a hyperplane and a related second hyperplane one obtains a nice geometry for the group stabilising the ovoidal hyperplane. We show that this group acts flag-transitively and that the geometry is simply connected. [ABSTRACT FROM AUTHOR]

Combining theoretical arguments with calculations in the computer algebra package GAP, we are able to show that there are 27 isomorphism classes of hyperplanes in the near hexagon for the group U(3). We give an explicit construction of a representative of each class and we list several combinatorial properties of such a representative. [ABSTRACT FROM AUTHOR]

Let π be a thick polar space of rank 4 over a field  such that the generalised quadrangle Res+(α) of a line α of π which consists of the planes and 3-spaces of π containing α, admits a spread. Let be a line-spread of π with the following property: Let be the set of 3-spaces of π in which induces spreads. For every point Σ of π, the 3-spaces of containing Σ all contain the spread line λ ∈ covering Σ and form a spread of the generalised quadrangle . Then is a generalised quadrangle. The polar spaces Sp8 () and admit such spreads for both finite and infinite fields . They are the only finite classical polar spaces admitting such spreads. If π ≅ Sp8( q), respectively , then Γ ≅ Sp4( q2), respectively H5( q2). If π ≅ Sp8() for some infinite field , then Γ ≅ Sp4() for some field . In the infinite case, there exists an example of a spread in H8(ℂ) over the complex numbers ℂ with over the quaternions. Dualizing π, the point set of the dual polar space Δ dual to π is a hyperplane of Δ intersecting each symp Σ, i.e. an element of maximal type of Δ, in the set of neighbours of an ovoid of a quad of Σ. [ABSTRACT FROM AUTHOR]

We apply diagram geometry and amalgam techniques to give a new proof of a theorem of K.-W. Phan, characterizing the special unitary group as a group generated by certain systems of subgroups SU (2,q2). [ABSTRACT FROM AUTHOR]

It is shown under minor extra assumptions that a P-geometry which has as its point residue the rank three P-geometry for the group M22 is the rank four P-geometry which has Co2 as its group of automorphisms. [ABSTRACT FROM AUTHOR]

It is shown under minor extra assumptions that a P-geometry which has as its point residue the rank three P-geometry for the group M22 is the rank four P-geometry which has Co2 as its group of automorphisms. [ABSTRACT FROM AUTHOR]

Hurwitz spaces are moduli spaces of curve covers. The isomorphism classes of covers of P¹ℂ with given ramification data are parameterized combinatorially by Nielsen tuples in the monodromy group G. The Artin braid group acts on Nielsen tuples, and the orbits of this action correspond to the connected components of the corresponding Hurwitz space. In this article we consider the case G = A5. We give a complete classification of the braid orbits for all ramification types, showing that the components are always distinguishable by the Fried-Serre lift invariant. [ABSTRACT FROM AUTHOR]

A connected 3-valent plane graph, whose faces are q- or 6-gons only, is called a graph qn. We classify all graphs 4n, which are isometric subgraphs of a m-hypercube Hm. [ABSTRACT FROM AUTHOR]