| ← Ch. 6: Groups | Table of contents | Ch. 8: Rings → |
This chapter proves theorems about an arbitrary group. We are given some
Grp : Group G for an unknown G, and we use only the fields from
Chapter 6. The point of this chapter lies less in the three theorems
themselves (they are standard) than in the search process for finding
each proof: given a goal, what to examine, what to attempt, and how to
recognize being stuck versus one step away. Each theorem below is presented
as that search, not merely its answer.
Learning objectives. By the end of this chapter, prove a fact about
every group at once by reasoning only from Group’s fields, recognize
the “relate both sides to a common third expression” and “pad with the
identity, then cancel” patterns, and reuse a proved lemma
(left_inverse_unique) to shortcut a later one (inv_op) instead of
re-deriving it.
Sections
- Setup
- Theorem 1: the identity is unique
- Theorem 2: left inverses equal right inverses
- Theorem 3: inverse of a product
- Exercises
| ← Ch. 6: Groups | Table of contents | Ch. 8: Rings → |