Let ξ : X j → E p → B be a fibration of simply connected CW complexes of finite type with classifying map h : B → Baut1(X). We study the evaluation subgroup Gn(E,X; j) of the fibre inclusion as an invariant of the fibre-homotopy type of ξ. For spherical fibrations, we show the evaluation subgroup may be expressed as an extension of the Gottlieb group of the fibre sphere provided the classifying...

The author has computed the class groups of all complex quadratic number fields Q(\f^~D) °f discriminant D for 0 < D < 4000000. In so doing, it was found that the first occurrences of rank three in the 3-Sylow subgroup are D = 3321607 = prime, class group C(3) x C(3) x C(9.7) (C(n) a cyclic group of order n), and D = 3640387 = 421.8647, class group C(3) X C(3) X C(9.2). The author has also foun...

The totality formed by all the operators of any group (G) which are common to all the invariant subgroups of prime index (p) constitutes a characteristic subgroup, and the corresponding quotient group is the abelian group of order pK and of type (1, 1, 1, ■■■)-\ The number of the invariant subgroups of index p is therefore pK — 1/p — 1. The given totality includes all the operators of G which a...

The totality formed by all the operators of any group (G) which are common to all the invariant subgroups of prime index (p) constitutes a characteristic subgroup, and the corresponding quotient group is the abelian group of order pK and of type (1, 1, 1, ■■■)-\ The number of the invariant subgroups of index p is therefore pK — 1/p — 1. The given totality includes all the operators of G which a...

Here we classify finite non-Dedekindian p-groups which are not generated by their non-normal subgroups. (Theorem 1). The purpose of this paper is to classify non-Dedekindian finite p-groups which are not generated by their non-normal subgroups. It is surprising that such p-groups must be of class 2 with a cyclic commutator subgroup. We consider here only finite p-groups and our notation is stan...

The abelian HSP Recall that in the hidden subgroup problem, we are given a function f : G→ S (for a known group G and a finite set S) satisfying f(x) = f(y) iff x and y are in the same (left) coset of the hidden subgroup H ≤ G. In this lecture we will use additive notation for the group operation of an abelian group, so we have f(x) = f(x) iff x − y ∈ H. The strategy for the general abelian HSP...

Let G be a finite group. The order supergraph of is the graph with vertex set G, and two distinct vertices x, y are adjacent if o(x)|o(y) or o(y)|o(x). enhanced power whose they generate cyclic subgroup. reduced 〈x〉⊂〈y〉 〈y〉⊂〈x〉. In this article, we characterize strong metric dimension supergraph,

1. At least three subgroups of Pneumococcus Type II may be recognized by specific immune reactions. They have been called Subgroups II A, II B, and II X. 2. That the organisms of these three subgroups are biologically related to Pneumococcus Type II is shown by the following facts: (a) Agglutination with Antipneumococcus Serum II. (b) Protection with Antipneumococcus Serum II, except Subgroup I...

Let x be an element of prime order, p, in a finite group, G, and let P be a p-Sylow subgroup of G containing x. Say that x satisfies the unique conjugacy condition (u.c.c.) relative to G and the p-Sylow subgroup P if x is not conjugate in G to any other element of P. Let 02'(G) denote the largest normal subgroup of G oi odd order. Define the subgroup Z*(G) by the equation Z*(G)/0v(G)=Z(G/0v(G))...