1,720,986 research outputs found
Quotients of maximal class of thin Lie algebras. The odd characteristic case
No full textAmong thin graded Lie algebras, which are particular instances of Lie algebras of finite width, there are many interesting objects, such as the graded Lie algebra associated to the Nottingham group. Among the factors of a thin algebra with respect to the terms of the lower central series, there is a greatest factor which is of maximal class. In thin Lie algebras associated to groups, this factor is metabelian.
In this paper we show that the same holds in general, provided the characteristic of the underlying field is odd. In another paper by the second author it is shown that this is not the case for characteristic two
From endomorphisms to bi-skew braces, regular subgroups, the Yang–Baxter equation, and Hopf–Galois structures
No full textThe interplay between set-theoretic solutions of the Yang–Baxter equation of Mathematical Physics, skew braces, regular subgroups, and Hopf–Galois structures has spawned a considerable body of literature in recent years.
In a recent paper, Alan Koch generalised a construction of Lindsay N. Childs, showing how one can obtain bi-skew braces (G,⋅,∘) from an endomorphism of a group (G,⋅) whose image is abelian.
In this paper, we characterise the endomorphisms of a group (G,⋅) for which Koch's construction, and a variation on it, yield (bi-)skew braces. We show how the set-theoretic solutions of the Yang–Baxter equation derived by Koch's construction carry over to our more general situation, and discuss the related Hopf–Galois structures
Skew braces from Rota-Baxter operators: A cohomological characterisation and some examples
Full text linkRota-Baxter operators for groups were recently introduced by L. Guo, H. Lang,
and Y. Sheng.
V. G. Bardakov and V. Gubarev showed that with each Rota-Baxter operator one
can associate a skew brace. Skew braces on a group can be characterised in
terms of certain gamma functions from to its automorphism group
, that are defined by a functional equation. For the
skew braces obtained from a Rota-Baxter operator the corresponding gamma
functions take values in the inner automorphism group
of .
In this paper, we give a characterisation of the gamma functions on a group
, with values in , that come from a Rota-Baxter
operator, in terms of the vanishing of a certain element in a suitable second
cohomology group.
Exploiting this characterisation, we are able to exhibit examples of skew
braces whose corresponding gamma functions take values in the inner
automorphism group, but cannot be obtained from a Rota-Baxter operator.
For gamma functions that can be obtained from a Rota-Baxter operators, we
show how to get the latter from the former, exploiting the knowledge that a
suitable central group extension splits.Comment: 14 pages; accepted for publication, Annali di Matematica Pura e
Applicat
Endomorphisms of 2-generated metabelian groups that induce the identity modulo the derived subgroup
No full textBrace blocks from bilinear maps and liftings of endomorphisms
Full text linkWe extend two constructions of Alan Koch, exhibiting methods to construct
brace blocks, that is, families of group operations on a set such that any
two of them induce a skew brace structure on .
We construct these operations by using bilinear maps and liftings of
endomorphisms of quotient groups with respect to a central subgroup.
We provide several examples of the construction, showing that there are brace
blocks which consist of distinct operations of any given cardinality.
One of the examples we give yields an answer to a question of Cornelius
Greither. This example exhibits a sequence of distinct operations on the
-adic Heisenberg group such that any two operations give a skew
brace structure on and the sequence of operations converges to the original
operation "".Comment: 19 page
Hopf-Galois structures on extensions of degree p2q and skew braces of order p2q: The cyclic Sylow p-subgroup case
Full text linkLet p,q be distinct primes, with p>2. We classify the Hopf-Galois structures on Galois extensions of degree p2q, such that the Sylow p-subgroups of the Galois group are cyclic. This we do, according to Greither and Pareigis, and Byott, by classifying the regular subgroups of the holomorphs of the groups (G,⋅) of order p2q, in the case when the Sylow p-subgroups of G are cyclic. This is equivalent to classifying the skew braces (G,⋅,∘). Furthermore, we prove that if G and Γ are groups of order p2q with non-isomorphic Sylow p-subgroups, then there are no regular subgroups of the holomorph of G which are isomorphic to Γ. Equivalently, a Galois extension with Galois group Γ has no Hopf-Galois structures of type G. Our method relies on the alternate brace operation ∘ on G, which we use mainly indirectly, that is, in terms of the functions γ:G→Aut(G) defined by g↦(x↦(x∘g)⋅g−1). These functions are in one-to-one correspondence with the regular subgroups of the holomorph of G, and are characterised by the functional equation γ(gγ(h)⋅h)=γ(g)γ(h), for g,h∈G. We develop methods to deal with these functions, with the aim of making their enumeration easier, and more conceptual
THE AUTOMORPHISM GROUPS OF GROUPS OF ORDER p(2)q
No full textWe record for reference a detailed description of the automorphism groups of the groups of order p(2)q, where p and q are distinct primes
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