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On Covering Groups With Proper Subgroups, Collin B. Moore

MSU Graduate Theses

In this paper, we explore groups that can be expressed as a union of proper subgroups. Using “covering number” to denote the minimal number of proper subgroups required to cover a group, we explore the nature of groups with covering numbers 3 and 4, while also finding covering numbers for p-groups, dihedral, and generalized dihedral groups.

Finite Groups In Which The Number Of Cyclic Subgroups Is 3/4 The Order Of The Group, James Alexander Cayley

MSU Graduate Theses

Let $G$ be a finite group, c(G) denotes the number of cyclic subgroups of G and α(G) = c(G)/|G|. In this thesis we go over some basic properties of alpha, calculate alpha for some families of groups, with an emphasis on groups with α(G) = 3/4, as all groups with α(G) > 3/4 have been classified by Garonzi and Lima (2018). We find all Dihedral group with this property, show all groups with α(G) = 3/4 have at least |G|/2-1 involutions, and discuss existing work by Wall (1970) and Miller (1919) classifying all such groups.

On The Hamiltonicity Of Subgroup Lattices, Nicholas Charles Fleece

MSU Graduate Theses

In this paper we discuss the Hamiltonicity of the subgroup lattices of

different classes of groups. We provide sufficient conditions for the

Hamiltonicity of the subgroup lattices of cube-free abelian groups. We also

prove the non-Hamiltonicity of the subgroup lattices of dihedral and

dicyclic groups. We disprove a conjecture on non-abelian p-groups by

producing an infinite family of non-abelian p-groups with Hamiltonian

subgroup lattices. Finally, we provide a list of the Hamiltonicity of the

subgroup lattices of every finite group up to order 35 barring two groups.

Groups Satisfying The Converse To Lagrange's Theorem, Jonah N. Henry

MSU Graduate Theses

Lagrange’s theorem, which is taught early on in group theory courses, states that the order of a subgroup must divide the order of the group which contains it. In this thesis, we consider the converse to this statement. A group satisfying the converse to Lagrange’s theorem is called a CLT group. We begin with results that help show that a group is CLT, and explore basic CLT groups with examples. We then give the conditions to guarantee either CLT is satisfied or a non-CLT group exists for more advanced cases. Additionally, we show that CLT groups are properly contained between …

When There Is A Unique Group Of A Given Order And Related Results, Haya Ibrahim Binjedaen

MSU Graduate Theses

It is well-known that any group whose order is a prime number must be cyclic, that is there is only one group of that order up to isomorphism. This is also the case for some non-prime orders, for example there is only one group of order 15 up to isomorphism. This thesis provides the necessary background material to completely characterize those n for which these is a unique group of order n, namely when n and the Euler phi function of n are relatively prime. We also determine for which n there are exactly two groups of order n up …