For an abstract of the article click on the title.



Kirtland, J., The Geometry of Finite Groups
Through the Use of Splitting Systems,
Communications in Algebra, 22(3), 735 - 764, 1994.

 

Kirtland, J., Direct Products of Finite
Inseparable Groups, Archiv der Mathematik, 62, 289
- 291, 1994.

 

Kirtland, J., Finite Solvable Multiprimitive
Groups, Communications in Algebra, 23(1), 335 -
356, 1995.

 

Kirtland, J., Direct Products of Inseparable
Finite Groups, II, Communications in Algebra, 25(3), 243 -
246, 1997

 

Kappe, L.C. and Kirtland, J., Some Analogues of
the Frattini Subgroup, Algebra Coloquium 4(4), 419-
426, 1997.

 

Kappe, L.C. and Kirtland, J., Supplementation
in Groups, Glasgow Mathematical Journal 42,
37-50,2000.

 

Hoh, P.- S. and Kirtland, J. , An Emergentist
Model for Writing in Mathematics,
ERIC ED 453552, 453-552, 2001

 

Kirtland, J and Hoh, P.- S., Integrating Mathematics
and Writing Instruction, PRIMUS XII(1),
11-26, 2002.

 

Kirtland, J., Identification Numbers and Check
Digit Schemes, Classroom Resourse Materials services,
The Mathematical Association of America,
Washington DC, 2002

 

Abstract for The Geometry of Finite Groups Through the Use of Splitting Systems

The splitting systems of a finite group are used to induce a geometry associated with the group. The method generalizes the classical approach used to induce a geometry associated with a finite dimensional vector space and extends concepts related to the special and projective linear groups to arbitrary finite groups. Applications are made to finite solvable nC-groups and to the automorphism group of homocyclic abelian p-groups.

Abstract for Direct Products of Finite Inseparable Groups

A finite group is inseparable if it does not split over any proper nontrivial normal subgroup; that is, if it has no nontrivial semidirect decomposition. Let G be a nontrivial finite group. Then G = T1...Tn for nontrivial inseparable subgroups Ti, where for 1 <= i < n, the product T1...Ti is normalized and avoided by Ti+1. The ordered set {T1, ... , Tn} is called a splitting system for G of length n. If n = 1, then G is inseparable. As the dihedral group of order 8 indicates, a group may have splitting systems of different lengths. However if a group is a direct product of inseparable groups, an analog of the Remak-Krull-Schmidt Theorem can be obtained.

Abstract for Finite Solvable Multiprimitive Groups

The investigation of the structure of finite solvable multiprimitive groups through an induced geometry leads to a classification of finite solvable multiprimitive groups of up to derived length 3 and to a necessary and sufficient geometric condition for when a finite solvable nC-group is multiprimitive. An application is made to finite groups which have all their splitting systems equivalent.

Abstract for Direct Products of Inseparable Finite Groups, II

A finite group is inseparable if it does not split over any proper nontrivial normal subgroup; that is, if it has no nontrivial semidirect decomposition. Let G be a nontrivial finite group. Then G = S1...Sn for nontrivial inseparable subgroups Si, where for 1 <= i < n, the product S1...Si is normalized and avoided by Si+1. This is denoted S1...Si+1 = [S1...Si](\theta}_{i})Si+1, where {\theta}_{i} is in Hom(Si+1 , Aut(S1...Si)) is induced through conjugation on the elements of $S1...Si by the elements in Si+1. The ordered set {S1, ... , Sn} is called a splitting system for G of length n. If for each i, 1 <= i <= n-1, {\theta}_{i} is the trivial map, then G = S1* ... *Sn is the direct product of nontrivial inseparable groups. Yet, the existence of such a decomposition does not always imply that all other splitting systems behave the same way. Necessary and sufficient conditions are established for when a finite group can be written as the direct product of nontrivial inseparable groups and for when all other decompositions give rise to such a direct product decomposition.

Abstract for Some Analogues of the Frattini Subgroup

For any group G, the Frattini subgroup Frat(G) is the intersection of the maximal subgroups of G. This paper investigates two other characteristic subgroups, nFrat(G) and cFrat(G), which are defined as the intersection of the maximal normal and maximal characteristic subgroups of G, respectively. The properties of nFrat(G) and cFrat(G) are studied and results analogous to those of the Frattini subgroup are established.

Abstract for Supplementation in Groups

In this paper, groups are investigated in which all subgroups, all normal subgroups,or all characteristic subgroups have a proper supplement.This supplement can be either an arbitrary subgroup,a normal or a characteristic subgroup,resulting in nine classes of groups.Properties of these are studied such as containment and closure properties, and characterizations for several of these classes are given.

Abstract for An Emergentist Model for Writing in Mathematics

This paper builds upon previous investigations of the relationship between textual features and cognitive processes in student writing. The present method involves extensive comparative analyses along multiple dimensions, using a large corpus of data from the revisions of freshmen in a writing-across-the-curriculum setting. The emergent patterns seen in the linguistic and quantitative data are interpreted within the larger context of findings in composition research, literacy studies, linguistics, writing acquisition research, and the historical study of languages. In providing empirical evidence for the writing-thinking connection, this study brings up the issue of evaluation.

Abstract for Integrating Mathematics and Writing Instruction

To counter students' sometimes fragmented approach to learning core skills, the authors have developed an integrated approach for teaching mathematics and writing to freshmen. The same students are enrolled in both an introductory level mathematics and a composition class. The goals are to strengthen mathematical skills, develop writing competencies, and foster interdisciplinary awareness. The foundation of this one semester two-course cluster is the creation of a unified learning environment in which writing is used to cultivate mathematical understanding and mathematics is used to develop writing skills.

Abstract for Identification Numbers and Check Digit Schemes

Identification numbers, used to encode information pertaining to products, documents, accounts, or individuals, are recorded onto documents, typed or scanned into computers, sent via the Internet, or transmitted in some other fashion millions of times a day. Given the heavy reliance on these numbers to transmit information and the possibility that a transmission error may occur, many add an extra digit or check digit that is used to determine if the identification number has been transmitted incorrectly. The mathematical process, called a check digit scheme, is used by the receiver of the number, independent of the sender, to recognize when a transmission error has occurred. This book presents the mathematics behind a variety of check digit schemes used today. Special attention is given to the airline ticket, United States Post Office money order, UPC, ISBN, IBM, and Verhoeff schemes. Topics from number theory, set theory and group theory are not only used to develop the schemes presented, but are used to develop topics from cryptography (RSA) and symmetry.

It may come as a surprise, but check digit schemes vary in their ability to catch errors. Some, such as the airline ticket scheme, do not catch every occurrence of the most common type of error, while others, such as the ISBN scheme, catch most error patterns. Consequently, the criteria used to judge the reliability of a scheme is a central theme of this book.

This book will be of interest to a wide audience, especially those interested in mathematics at work. It is an ideal text for a liberal arts mathematics class. The book is organized to allow students to move from simple mathematical concepts and check digit schemes to more complex ideas. It also provides writing and group activities, which can be integrated into a student-centered approach.

To the Marist Homepage | To the Mathmatics Department Homepage | Feedback | Back to my Welcome Page