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The centre of a Steiner loop and the maxi-Pasch problem

Andrew R. Kozlik (2020)

Commentationes Mathematicae Universitatis Carolinae

A binary operation “ · ” which satisfies the identities x · e = x , x · x = e , ( x · y ) · x = y and x · y = y · x is called a Steiner loop. This paper revisits the proof of the necessary and sufficient conditions for the existence of a Steiner loop of order n with centre of order m and discusses the connection of this problem to the question of the maximum number of Pasch configurations which can occur in a Steiner triple system (STS) of a given order. An STS which attains this maximum for a given order is said to be maxi-Pasch. We show that...

The classification of finite groups by using iteration digraphs

Uzma Ahmad, Muqadas Moeen (2016)

Czechoslovak Mathematical Journal

A digraph is associated with a finite group by utilizing the power map f : G G defined by f ( x ) = x k for all x G , where k is a fixed natural number. It is denoted by γ G ( n , k ) . In this paper, the generalized quaternion and 2 -groups are studied. The height structure is discussed for the generalized quaternion. The necessary and sufficient conditions on a power digraph of a 2 -group are determined for a 2 -group to be a generalized quaternion group. Further, the classification of two generated 2 -groups as abelian or non-abelian...

The cohomology algebras of orientable Seifert manifolds and applications to Lusternik-Schnirelmann category

J. Bryden, P. Zvengrowski (1998)

Banach Center Publications

This note gives a complete description of the cohomology algebra of any orientable Seifert manifold with ℤ/p coefficients, for an arbitrary prime p. As an application, the existence of a degree one map from an orientable Seifert manifold onto a lens space is completely determined. A second application shows that the Lusternik-Schnirelmann category for a large class of Seifert manifolds is equal to 3, which in turn is used to verify the Ganea conjecture for these Seifert manifolds.

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