How do undergraduate students cope with abstract algebra concepts? How should we go about researching this question? Based on interviews with undergraduate students and on written questionnaires, a theoretical framework evolved which could coherently account for most of the data. According to this theoretical framework, students' responses can be interpreted as a result of reducing the level of abstraction. In this paper, the theme of reducing abstraction is examined, based on three interpretations for levels of abstraction discussed in mathematics education research literature. From these three perspectives on abstraction, ways in which students reduce abstraction level are analyzed and exemplified.

Reducing Abstraction Level When Learning Abstract Algebra Concepts.

The influence of SS-quasinormality of some subgroups on the structure of finite groups☆

A subgroup H of a group G is said to be π-quasinormal in G if it permutes with every Sylow subgroup of G, and H is said to be π-quasinormally embedded in G if for each prime dividing the order of H, a Sylow p-subgroup of H is also a Sylow p-subgroup of some π-quasinormal subgroups of G. We characterize p-nilpotentcy of finite groups with the assumption that some maximal subgroups, 2-maximal subgroups, minimal subgroups and 2-minimal subgroups are π-quasinormally embedded, respectively.

On p-nilpotency of finite groups with some subgroups π-quasinormally embedded

Categorically compact topological groups

A brief survey of the development of the theory of closure operators is presented Results concerning the applications of the theory to epimorphisms, separation, compactness and connectedness are also included together with a number of supporting examples

Categorical closure operators

Trivial intersection groups

This paper examines group automorphisms for which each element commutes with its image. These automorphisms do not necessarily form a subgroup of the automorphism group, but have a number of interesting properties, close to those of central automorphisms.

On commuting automorphisms of groups

The power graph of a finite group is the graph whose vertex set is the group, two distinct elements being adjacent if one is a power of the other. In this article, we classify the finite gr...

Power graphs of (non)orientable genus two

This paper characterizes those finite groups with the property that maximal subgroups of Sylow subgroups are normal. They are all certain extensions of nilpotent groups by cyclic groups.

Groups with maximal subgroups of Sylow subgroups normal

Let $\Gamma$ be a graph with vertex set $V(\Gamma)$. A subset $C$ of $V(\Gamma)$ is called a perfect code in $\Gamma$ if $C$ is an independent set of $\Gamma$ and every vertex in $V(\Gamma)\setminus C$ is adjacent to exactly one vertex in $C$. A subset $C$ of a group $G$ is called a perfect code of $G$ if there exists a Cayley graph of $G$ which admits $C$ as a perfect code. A group $G$ is said to be code-perfect if every proper subgroup of $G$ is a perfect code of $G$. In this paper we prove that a group is code-perfect if and only if it has no elements of order $4$. We also prove that a proper subgroup $H$ of an abelian group $G$ is a perfect code of $G$ if and only if the Sylow $2$-subgroup of $H$ is a perfect code of the Sylow $2$-subgroup of $G$. This reduces the problem of determining when a given subgroup of an abelian group is a perfect code to the case of abelian $2$-groups. Finally, we determine all subgroup perfect codes in any generalized quaternion group.

Gary L. Walls

Papers

Trivial intersection groups

On commuting automorphisms of groups

Power graphs of (non)orientable genus two

Groups with maximal subgroups of Sylow subgroups normal

Subgroup perfect codes in Cayley graphs