A Generalization of Hall's Theorem: 10701

Let [Formula: see text] be a finite group, whose order has [Formula: see text] prime divisors. In this paper, we prove that if [Formula: see text] has a [Formula: see text]-complement for [Formula: see text] prime divisors [Formula: see text] of [Formula: see text] and [Formula: see text] has no section isomorphic to [Formula: see text]. Then [Formula: see text] is solvable, which generalizes a theorem of Hall.

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Hall's theorem and extending partial latinized rectangles

Journal of Combinatorial Theory Series A ◽

10.1016/j.jcta.2014.10.007 ◽

2015 ◽

Vol 130 ◽

pp. 26-41

Author(s):

J.L. Goldwasser ◽

A.J.W. Hilton ◽

D.G. Hoffman ◽

Sibel Özkan

Keyword(s):

Hall's Theorem ◽

Hall’S Theorem

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Symmetric polynomials and Hall's theorem

Discrete Mathematics ◽

10.1016/0012-365x(88)90051-9 ◽

1988 ◽

Vol 69 (3) ◽

pp. 225-234 ◽

Cited By ~ 3

Author(s):

Klaus G. Fischer

Keyword(s):

Symmetric Polynomials ◽

Hall's Theorem ◽

Hall’S Theorem

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Factoring Graphs

The Fascinating World of Graph Theory ◽

10.23943/princeton/9780691175638.003.0007 ◽

2017 ◽

Author(s):

Arthur Benjamin ◽

Gary Chartrand ◽

Ping Zhang

Keyword(s):

Graph Theory ◽

Bipartite Graphs ◽

Petersen Graph ◽

Different Types ◽

Bridgeless Graph ◽

Hall's Theorem ◽

Hall’S Theorem

This chapter focuses on Hall's Theorem, introduced by British mathematician Philip Hall, and its connection to graph theory. It first considers problems that ask whether some collection of objects can be matched in some way to another collection of objects, with particular emphasis on how different types of schedulings are possible using a graph. It then examines one popular version of Hall's work, a statement known as the Marriage Theorem, the occurrence of matchings in bipartite graphs, Tutte's Theorem, Petersen's Theorem, and the Petersen graph. Peter Christian Julius Petersen introduced the Petersen graph to show that a cubic bridgeless graph need not be 1-factorable. The chapter concludes with an analysis of 1-factorable graphs, the 1-Factorization Conjecture, and 2-factorable graphs.

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