Homotopy of braids on surfaces: Extending Goldsmith’s answer to Artin

In 1947, in the paper “Theory of Braids,” Artin raised the question of whether isotopy and homotopy of braids on the disk coincide. Twenty seven years later, Goldsmith answered his question: she proved that in fact the group structures are different, exhibiting a group presentation and showing that the homotopy braid group on the disk is a proper quotient of the Artin braid group on the disk [Formula: see text], denoted by [Formula: see text]. In this paper, we extend Goldsmith’s answer to Artin for closed, connected and orientable surfaces different from the sphere. More specifically, we define the notion of homotopy generalized string links on surfaces, which form a group which is a proper quotient of the braid group on a surface [Formula: see text], denoting it by [Formula: see text]. We then give a presentation of the group [Formula: see text] and find that the Goldsmith presentation is a particular case of our main result, when we consider the surface [Formula: see text] to be the disk. We close with a brief discussion surrounding the importance of having such a fixed construction available in the literature.

Moufang loops that are almost groups

It is known that Moufang loops are closely related to groups as they have many properties in common. For instance, they both have an inverse for each element and satisfy Lagrange's theorem. In this paper, we study the properties of a class of Moufang loops which are not groups, but all their proper subloops and proper quotient loops are groups.

A Survey on Just-Non-𝔛Groups

Let𝔛be a class of groups. A group which does not belong to𝔛but all of whose proper quotient groups belong to𝔛is called just-non-𝔛group. The present note is a survey of recent results on the topic with a special attention to topological groups.

Generalized Verma Modules over Lie Algebras of Weyl Type

For a field 𝔽 of characteristic 0 and an additive subgroup Γ of 𝔽, there corresponds a Lie algebra [Formula: see text] of generalized Weyl type. Given a total order of Γ and a weight Λ, a generalized Verma [Formula: see text]-module M(Λ, ≺) is defined. In this paper, the irreducibility of M(Λ, ≺) is completely determined. It is also proved that an irreducible highest weight module over the [Formula: see text]-infinity algebra [Formula: see text] is quasifinite if and only if it is a proper quotient of a Verma module.

Verma Modules over a Block Lie Algebra

Let [Formula: see text] be the Lie algebra with basis {Li,j, C|i, j ∈ ℤ} and relations [Li,j, Lk,l] = ((j + 1)k - i(l + 1))Li+k, j+l + iδi, -kδj+l, -2C and [C, Li,j] = 0. It is proved that an irreducible highest weight [Formula: see text]-module is quasifinite if and only if it is a proper quotient of a Verma module. An additive subgroup Γ of 𝔽 corresponds to a Lie algebra [Formula: see text] of Block type. Given a total order ≻ on Γ and a weight Λ, a Verma [Formula: see text]-module M(Λ, ≻) is defined. The irreducibility of M(Λ, ≻) is completely determined.

Finite groups that need more generators than any proper quotient

A structure theorem is proved for finite groups with the property that, for some integer m with m ≥ 2, every proper quotient group can be generated by m elements but the group itself cannot.