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
• Equality of left and right neutral element
• Equality of left and right inverses
• Equivalence of definitions of group
• Invertible implies cancellative
• Equivalence of definitions of subgroup
• Inverse map is involutive
• Associative binary operation
• Finite group
• Subsemigroup of finite group is subgroup
• Sufficiency of subgroup criterion
• Manipulating equations in groups

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

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