Tour:Introduction two (beginners)

This page is part of the Groupprops Guided tour for beginners (Jump to beginning of tour)
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Hopefully, by the time you've reached this part of the guided tour, you have the basic definitions of a group and have some understanding of this definition. In this part, we'll see more about how to prove simple things about groups, and how to manipulate equations in groups.

This part focuses on providing an understanding of how to do simple manipulations involving groups. We begin by generalizing some of the ideas involving groups, and discussing proofs involving some of the basic manipulations.

We'll see the following pages:

• Some variations of group: Defines weaker notions than groups, where one or more of the axioms or conditions for a group is relaxed.
• Equality of left and right neutral element: A short, mind's eye proof.
• Equality of left and right inverses: A short, mind's eye proof.
• Equivalence of definitions of group: Consolidates the definition of group by proving how two apparently different definitions seen in part one are equivalent.
• Invertible implies cancellative: A short, mind's eye proof.
• Equivalence of definitions of subgroup: Consolidates the definition of group by proving how two apparently different definitions seen in part one are equivalent.
• Associative binary operation: Defines and discusses important aspects of associative binary operations.
• Inverse map is involutive: A short, mind's eye proof.
• Order of a group: A simple definition.
• Finite group: A simple definition.
• Subsemigroup of finite group is subgroup: Applies ideas seen previously to prove a simple result on subsets of finite groups.
• Sufficiency of subgroup criterion: Applies ideas seen previously to prove a certain criterion for checking whether a subset is a subgroup.
• Manipulating equations in groups: Explores how equations and expressions in groups are manipulated. Consolidates material seen in parts one and two.

We'll also see some consolidation pages:

• Factsheet two: Summarizes facts seen in parts one and two.
• Confidence aggregator two: Asks questions to help the reader self-assess and introspect on what has been learned in parts one and two.
• Mind's eye test two: Problems based on parts one and two, that help sharpen the mind's eye and consolidate material learned in these parts.

Prerequisites, goal and general suggestions

Prerequisites for this part: Material covered in part one, or equivalent. Basically, the definitions of group, subgroup, trivial group and Abelian group.

Desirables for this part: Experience with a few groups, like the additive group of real or rational numbers, and with some monoids, such as the additive monoid of nonnegative integers (knowing the abstract concepts of group and monoid isn't necessary).

The goal of this part is to:

• Provide some intuition into how to manipulate the various conditions for being a group, to prove simple statements about groups
• Give an idea of the way the axioms control and make rigid the structure of a group