A Summary of Inquiry-Based Approach to Abstract Algebra by Dana C. Ernst

An Inquiry-Based Approach to Abstract Algebra

General Information

Authored by Dana C. Ernst, PhD, Northern Arizona University, Fall 2021.
The material is designed as a task sequence for an undergraduate abstract algebra course using an inquiry-based learning (IBL) approach.
The latest version is available on GitHub: http://dcernst.github.io/IBL-AbstractAlgebra/
The work is licensed under the Creative Commons Attribution-Share Alike 4.0 United States License.
Attribution: Dana C. Ernst, Mathematics Faculty at Northern Arizona University, dana.ernst@nau.edu.
License details: https://creativecommons.org/licenses/by-sa/4.0/

Contents Overview

Preface: Focuses on ideas over calculations, mathematical discovery, rigorous thinking, beauty, elegance, and aesthetic value in mathematics. Stresses producing rather than consuming knowledge and developing independent, inquisitive, and persistent problem solvers.
Acknowledgements: Lists contributors who provided feedback or content.
Chapter 1: Introduction
- 1.1 What is Abstract Algebra?: Defines abstract algebra as the study of algebraic structures like groups, rings, fields, modules, vector spaces, and algebras, with a focus on group theory.
- 1.2 What Should You Expect?: Highlights the shift from using formulas to mathematical experimentation, conjecture, definition writing, and theorem proving. Emphasizes producing and discovering mathematics through wrestling with mathematical ideas.
- 1.3 An Inquiry-Based Approach: Describes inquiry-based learning (IBL) as a student-centered method engaging students in sense-making activities and creating or discovering mathematics.
  - IBL involves solving problems, conjecturing, experimenting, exploring, creating, and communicating.
  - Four Pillars of IBL:
    - Deep engagement with meaningful tasks.
    - Collaborative processing of mathematical ideas.
    - Instructor inquiry into student thinking.
    - Promotion of equity in design and facilitation.
- 1.4 Structure of the Textbook: Explains the use of definitions, examples, problems, theorems, and corollaries to promote active engagement and discovery.
- 1.5 Some Minimal Guidance: Provides guidelines for writing mathematical proofs, including stating assumptions clearly, writing in complete sentences, referencing definitions, and indicating the beginning and end of proofs.
Chapter 2: An Introduction to Groups
- 2.1 A First Example: Introduces group theory using the game Spinpossible™ on a 3x3 board.
 - $s_{ij}$ denotes the spin that rotates the subrectangle with position $i$ in the upper-left corner and position $j$ in the lower-right corner.
 - The primary observations about $Spin_{3 \times 3}$ :
 - The set of spins generates $Spin_{3 \times 3}$ .
 - The composition of any two net actions from $Spin{3 \times 3}$ results in a net action from $Spin{3 \times 3}$ .
 - The composition of net actions from $Spin_{3 \times 3}$ is associative.
 - There is an identity in $Spin_{3 \times 3}$ whose corresponding net action is “do nothing.”
 - Every net action from $Spin{3 \times 3}$ has an inverse net action in $Spin{3 \times 3}$ . Composing a net action and its inverse results in the identity.
 - The composition of two net actions from $Spin_{3 \times 3}$ may or may not commute.
- 2.2 Binary Operations: Defines a binary operation on a set $A$ as a function from $A \times A$ into $A$ .
- 2.3 Groups: Definition of a group $(G,*)$ , axioms, and key properties.
 - Axioms include:
 - Closure under $\ast$ .
 - Associativity of $\ast$ .
 - Existence of an identity element $e$ .
 - Existence of inverses for all elements.
 - The order of $G$ , denoted $|G|$ , is the cardinality of the set $G$ .
 - If $\ast$ is commutative, then $G$ is abelian.
 - Theorem 2.37: If $G$ is a group, then there is a unique identity element in $G$ .
 - Theorem 2.39 (Cancellation Law): Let $G$ be a group and let $g, x,y \in G$ . Then $gx = gy$ if and only if $x = y$
 - Theorem 2.41: If $G$ is a group, then each $g \in G$ has a unique inverse.
 - Theorem 2.44: If $G$ is a group, then $(gh)^{-1} = h^{-1}g^{-1}$ for all $g, h \in G$ .
 - Definition 2.45: introduces multiplicative notation for integer powers of a group element $g$ including negative powers and $g^0=e$ .
 - Theorem 2.47: If $G$ is a group and $g \in G$ , then for all $n,m \in \mathbb{Z}$ , we have:
 - $g^ng^m = g^{n+m}$ ,
 - $(g^n)^{-1} = g^{-n}$ ,
 - $(g^n)^m = g^{nm}$ .
- 2.4 Generating Sets: Introduces generating sets and cyclic groups. If $G$ is a group and $S \subseteq G$ , then $\langle S \rangle$ is a subgroup of $G$ , called the subgroup generated by $S$ .
 - If $\langle S \rangle = G$ , then $S$ is called a generating set of $G$ .
 - A generating set $S$ for $G$ is a minimal generating set if $S \setminus {x}$ is no longer a generating set for $G$ for all $x \in S$ .
- 2.5 Group Tables: Discusses representing finite groups using group tables.
 - Theorem 2.65: If $(G,*)$ is a finite group, then each element of $G$ appears exactly once in each row and each column, respectively, in any group table for $G$ .
- 2.6 Cayley Diagrams: Explains visual encoding of a group's structure using a specified generating set.
 - Definition 2.70: Suppose $G$ is a group and $S$ is a generating set of $G$ . The Cayley diagram for $G$ with generating set $S$ is a colored directed graph constructed as follows:
 - The vertices correspond to elements of $G$ .
 - Each generator $s \in S$ is assigned a color, say $c_s$ .
 - For $g \in G$ and $s \in S$ , there is a directed edge from $g$ to $sg$ with color $c_s$ .
 - Theorem 2.76: If $G$ is a group with generating set $S$ , then for every $g \in G$ and $s \in S$ , there is exactly one arrow with color $cs$ pointing from $s^{-1}g$ to $g$ and exactly one arrow with color $cs$ pointing from $g$ to $sg$ .
 - Theorem 2.77: If $G$ is a group with generating set $S$ , then the Cayley diagram for $G$ with generating set $S$ is connected.
Chapter 3: Subgroups and Isomorphisms
- 3.1 Subgroups: Defines subgroups and provides the Two-Step Subgroup Test (Theorem 3.6), which states that a nonempty subset H of a group G is a subgroup if and only if it is closed under inverses and under the binary operation of G.
 - Theorem 3.7: If $G$ is a group, then ${e} \le G$ .
 - Theorem 3.9: If $G$ is a group, then $G \le G$ .
 - Theorem 3.10: If $G$ is a group and $S \subseteq G$ , then $\langle S \rangle \le G$ .
 - Theorem 3.19: If $G$ is an abelian group such that $H \le G$ , then $H$ is an abelian subgroup.
 - Definition 3.20: If $G$ is a group, then we define the center of $G$ to be
 $Z(G) := {z \in G \mid zg = gz \text{ for all } g \in G }$ .
 - Theorem 3.21: If $G$ is a group, then $Z(G)$ is an abelian subgroup of $G$ .
- 3.2 Subgroup Lattices: Introduces subgroup lattices and related concepts.
 - Theorem 3.23: Let $G$ be a group and let $g1,g2,…,gn \in G$ . If $x \in \langle g1,g2,…,gn \rangle$ , then $\langle g1,g2,…,gn \rangle = \langle g1,g2,…,gn, x \rangle$ .
 - Theorem 3.24: If $G$ is a group such that $H,K \le G$ , then $H \cap K \le G$ . Moreover, $H \cap K$ is the largest subgroup contained in both $H$ and $K$ .
 - Theorem 3.26: If $G$ is a group such that $H,K \le G$ , then $\langle H \cup K \rangle \le G$ . Moreover, $\langle H \cup K \rangle \le G$ is the smallest subgroup containing both $H$ and $K$ .
- 3.3 Isomorphisms: Formalizes the notion of when two groups have the same structure.
 - Theorem 3.34: If there is a matching between $G1$ and $G2$ using the generating sets $T1$ and $T2$ , respectively, then $|G1| = |G2|$ and $T1$ and $T2$ have the same cardinality.
 - Theorem 3.45: If $G1$ and $G2$ are two finite groups, then there is a matching between $G1$ and $G2$ if and only if $G1$ and $G2$ have an identical table coloring.