Let G be a finite group multiplicatively written. The small Davenport constant of G is the maximum positive integer d(G) such that there exists a product-one free sequence S of length d(G). Let ...s2≡1(modn), where s≢±1(modn). It has been proven that d(Cn⋊sC2)=n (see 13, Lemma 6). In this paper, we determine all sequences over Cn⋊sC2 of length n which are product-one free. It completes the classification of all product-one free sequences over every group of the form Cn⋊sC2, including the quasidihedral groups and the modular maximal-cyclic groups.
Full text
Available for:
GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
Abstract
The paper is devoted to projective Clifford groups of quantum
N
-dimensional systems (with configuration space
Z
N
). Clearly, Clifford gates allow only the simplest quantum computations ...which can be simulated on a classical computer (Gottesmann–Knill theorem). However, it may serve as a cornerstone of full quantum computation. As to its group structure it is well-known that—in
N
-dimensional quantum mechanics—the Clifford group is a natural semidirect product provided the dimension
N
is an odd number. For even
N
special results on the Clifford groups are scattered in the mathematical literature, but they mostly do not concern the semidirect structure. Using appropriate group presentation of
S
L
(
2
,
Z
N
)
it is proved that for even
N
the projective Clifford groups are not natural semidirect products if and only if
N
is divisible by four.
Tropical linear algebra has been recently put forward by Grigoriev and Shpilrain as a promising platform for implementation of protocols of Diffie-Hellman and Stickel type. Based on the CSR expansion ...of tropical matrix powers, we suggest a simple algorithm for the following tropical discrete logarithm problem: "Given that
for a unique t and matrices A, V, F of appropriate dimensions, find this t." We then use this algorithm to suggest a simple attack on a protocol based on the tropical semidirect product. The algorithm and the attack are guaranteed to work in some important special cases and are shown to be efficient in our numerical experiments.
Full text
Available for:
BFBNIB, GIS, IJS, KISLJ, NUK, PNG, UL, UM, UPUK
For a finite group G, and a subset S of G (possibly, contains the identity element), the bi-Cayley graph Bcay(G, S) of G with respect to S is defined as the undirected bipartite graph with vertex set
...and edge set
. In this paper, we give some sufficient conditions for the existence of hamiltonian cycle in
, where
is a semiproduct of Z
m
by a subgroup H of G. In addition, we introduce a new graph operation, which produces some classes of bi-Cayley graphs having hamiltonian cycle.
Full text
Available for:
BFBNIB, GIS, IJS, KISLJ, NUK, PNG, UL, UM, UPUK
We determine normal forms of the multiplication of four-dimensional anti-commutative algebras over a field K of characteristic zero having an analogous family of flags of subalgebras as the ...four-dimensional non-Lie binary Lie algebras, and hence can be considered as the closest relatives of binary Lie algebras. These algebras are extensions of K by the 3-dimensional nilpotent Lie algebra and at the same time extensions of a two-dimensional Lie algebra by a two-dimensional abelian algebra. We describe their groups of automorphisms as extensions of a subgroup of the group of automorphisms of the three-dimensional nilpotent Lie algebra by K.
Full text
Available for:
GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
In the study of pre-Lie algebras, the concept of pre-morphism arises naturally as a generalization of the standard notion of morphism. Pre-morphisms can be defined for arbitrary (not-necessarily ...associative) algebras over any commutative ring k with identity, and can be dualized in various ways to generalized morphisms (related to pre-Jordan algebras) and anti-pre-morphisms (related to anti-pre-Lie algebras). We consider idempotent pre-endomorphisms (generalized endomorphisms, anti-pre-endomorphisms). Idempotent pre-endomorphisms are related to semidirect-product decompositions of the sub-adjacent anticommutative algebra.
Full text
Available for:
BFBNIB, GIS, IJS, KISLJ, NUK, PNG, UL, UM, UPUK
Weakly Schreier split extensions are a reasonably large, yet well-understood class of monoid extensions, which generalise some aspects of split extensions of groups. This short note provides a way to ...define and study similar classes of split extensions in general algebraic structures (parameterised by a term
θ
). These generalise weakly Schreier extensions of monoids, as well as general extensions of semi-abelian varieties (using the
θ
appearing in their syntactic characterisation). Restricting again to the case of monoids, a different choice of
θ
leads to a new class of monoid extensions, more general than the weakly Schreier split extensions.
Full text
Available for:
EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
PDF
