We study bipartite distance-regular Cayley graphs with diameter three or four. give sufficient conditions under which a graph can be constructed on the semidirect product of group — part this contains identity element and $\mathbb{Z}_{2}$. apply to case three, consider cases where are not satisfied for some specific groups such as dihedral group.We also extend result by Miklavič Potočnik that r...

Introduction. In the following all semigroups are of finite order. One semigroup Si is said to divide another semigroup S2, written Si|S2, if Si is a homomorphic image of a subsemigroup of S2. The semidirect product of S2 by Si, with connecting homomorphism Y, is written S2 Xy Si. See Definition 1.6. A semigroup S is called irreducible if for all finite semigroups S2 and Si and all connecting h...

We classify, among the linear algebraic groups over a local field of characteristic zero, those that have the Haagerup property (also called a-(T)-menability). Our method relies essentially on a discussion on the existence of a subgroup isomorphic, up to a finite covering, to the semidirect product of SL2 by an irreducible representation, or a one-dimensional central extension of an even-dimens...

The braided approach to q-deformation (due to the author and collaborators) gives natural algebras R21u1Ru2 = u2R21u1R and R21x1x2 = x2x1R for q-Minkowski and q-Euclidean spaces respectively. These algebras are covariant under a corresponding background ‘rotation’ quantum group. Semidirect product by this according to the bosonisation procedure (also due to the author) gives the corresponding P...

The holomorph of a free group Fn is the semidirect product FnoAut(Fn). Using the methods of Hatcher and Vogtmann in [10] and [11], we derive stability results and calculate the mod-p homology of these holomorphs for odd primes p in dimensions 1 and 2, and their rational homology in dimensions 1 through 5. Calculations of the twisted (where Aut(Fn) acts by first projecting to Gln(Z) and then inc...

Let G be a group which admits the structure of an iterated product of central extensions and semidirect products of abelian groups Gi (both finite and infinite). We describe a Mathematica 4.0 notebook for computing the homology of G, in terms of some homological models for the factor groups Gi and the products involved. Computational results provided by our program have allowed the simplificati...

W. Magnus’ representations of submonoids E ≤ End(F ) of the endomorphisms of a free group F of finite rank are generalised by identifying them with the first homology group of F with particular coefficient modules. By considering a suitable free resolution of the integers over the semidirect product of free groups, a class of representations of the braid group can be obtained on higher homology...

Without the factor of 1 2 , this would be the semidirect product Lie algebra for the usual action of gl(n,R) on R. With the factor of 1 2 , the bracket does not satisfy the Jacobi identity. Nevertheless, it does satisfy the Jacobi identity on many subspaces which are closed under the bracket. In fact, we will see that any Lie algebra structure on R is realized on such a subspace. If B is any bi...

‎For a homogeneous spaces ‎$‎G/H‎$‎, we show that the convolution on $L^1(G/H)$ is the same as convolution on $L^1(K)$, where $G$ is semidirect product of a closed subgroup $H$ and a normal subgroup $K $ of ‎$‎G‎$‎. ‎Also we prove that there exists a one to one correspondence between nondegenerat $ast$-representations of $L^1(G/H)$ and representations of ...

We study the free objects in variety of semigroups and monoids generated by monoid all n×n upper triangular matrices over a commutative semiring. obtain explicit representations these, as multiplicative subsemigroups quiver algebras polynomial semirings. A case particular interest is where n=2 semiring tropical semifield, coincides with that bicyclic (or equivalently, monogenic inverse monoid);...