The purpose of this paper is to define a new equivalence relation τ∗ on divisible hypergroups and to show that this relation is the smallest strongly regular relation (the fundamental relation) on commutative divisible hypergroups. We show that τ∗ ̸= β∗, τ∗ ̸= γ∗ and, we define a divisible hypergroup on every nonempty set. We show that the quotient of a finite divisible hypergroup by τ∗ is the tr...

We give a C0051JUCtion for new families of semi-regular divisible difference sets. The construction iJ a variation of McFarland's scheme [Sl tor nonc;yclic difference SCIS. • Let G be a group of order 11111 and N a subgroup of G of order n. If D is a k-subset of G then Dis a {m, n, le , >.1, >.2) divisible difference set in G relative to N provided that the differences dd'1 fur d, d ' ED, d ;e ...

We study Tamagawa numbers of elliptic curves with torsion Z/2Z?Z/14Z over cubic fields and an n-isogeny Q, for n?{6,8,10,12,14,16,17,18,19,37,43,67,163}. Bruin Najman [3] proved that every curve a field is base change defined Q. find Q are always divisible by 142, each factor 14 coming from rational prime split multiplicative reduction type I14k, one which p=2. The only exception the 1922.e2, c...

Let p be a prime number and let n an integer not divisible by such that every group of order np has normal subgroup p. (This holds in particular for p>n.) Under these hypotheses, we obtain one-to-one correspondence between the isomorphism classes braces size set pairs (Bn,[τ]), where Bn runs over [τ] morphisms from multiplicative to Zp⁎ under certain equivalence relation. This gives classificat...

Let G be connected reductive algebraic group defined over an algebraically closed field of characteristic p>0 and suppose that p is a good prime for the root system G, derived subgroup simply Lie algebra g=Lie(G) admits non-degenerate (AdG)-invariant symmetric bilinear form. Given linear function ? on g we denote by U?(g) reduced enveloping associated with ?. By Kac–Weisfeiler conjecture (now t...

Previous formulations of group theory in ACL2 and Nqthm, based on either "encapsulate" or "defn-sk", have been limited by their failure to provide a path proof induction the order group, which is required for most interesting results this domain beyond Lagrange's Theorem (asserting divisibility that subgroup). We describe an alternative approach finite remedies deficiency, explicit representati...

We show that if G is a finite Abelian group and the block size is 3, then the necessary conditions for the existence of a (v, 3, λ;G) GBRD are sufficient. These necessary conditions include the usual necessary conditions for the existence of the associated (v, 3, λ) BIBD plus λ ≡ 0 (mod |G|), plus some extra conditions when |G| is even, namely that the number of blocks be divisible by 4 and, if...

Originally this was to be entirely on Geometry in computer algebra systems|especially CAYLEY/MAGMA and GAP (but perhaps also at least partially in other systems such as MAPLE)|to some extent just a wish list. However, at the request of John Cannon, that will be the second of two parts: I'll start with PART I. Some geometric algorithms (and applications). There are a couple of ways to phrase the...

in this paper, we extend the construction of a fuzzy subgroup generated by a fuzzy subset to $l$-setting. this construction is illustrated by an example. we also prove that for an $l$-subset of a group, the subgroup generated by its level subset is the level subset of the subgroup generated by that $l$-subset provided the given $l$-subset possesses sup-property.

Abstract The article is motivated by the recently published studies on injective and projective hypermodules. We present here a new characterization of normal First we define concept zero-divisors over hypermodule based it introduce class hypermodules, one divisible After presenting some their fundamental properties, will show that R -hypermodules M coincide whenever hyperring with no . Finally...