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On The Number Of Distinct Cyclic Subgroups Of A Given Finite Group, Joseph Dillstrom
On The Number Of Distinct Cyclic Subgroups Of A Given Finite Group, Joseph Dillstrom
MSU Graduate Theses
In the study of finite groups, it is a natural question to consider the number of distinct cyclic subgroups of a given finite group. Following an article by M. Tarnauceanu in the American Mathematical Monthly, we consider arithmetic relations between the order of a finite group and the number of its cyclic subgroups. We classify several infinite families of finite groups in this fashion and expand upon an open problem posed in the article.
A Cycle Generating Function On Finite Local Rings, Tristen Kirk Wentling
A Cycle Generating Function On Finite Local Rings, Tristen Kirk Wentling
MSU Graduate Theses
We say a function generates a cycle if its output returns the initial value for some number of successive applications of . In this thesis, we develop a class of polynomial functions for finite local rings and associated functions . We show that the zeros of one are precisely the fixed points of the other and that every ring element is either one of these fixed points or is in a cycle of fixed length equal to the order of 2 in the associated group of units. Particular emphasis is given to rings of integers modulo the square of a …