Abstract Algebra I - Chapter 6: Isomorphisms

Flashcards for Abstract Algebra I, Chapter 6: Isomorphisms, covering definitions, theorems, and exercises.

Isomorphism (Definition 1)

A one-to-one and onto mapping 𝜌 from 𝐺1 to 𝐺2 that preserves the group operation: 𝜌(𝑎 ∙ 𝑏) = 𝜌(𝑎) ∙ 𝜌(𝑏) for all 𝑎, 𝑏 ∈ 𝐺1. If such 𝜌 exists, 𝐺1 and 𝐺2 are isomorphic (𝐺1 ≃ 𝐺2).

Automorphism

An isomorphism from a group onto itself.

Steps to Prove 𝐺1 ≃ 𝐺2

1) Define a mapping 𝜌 from 𝐺1 to 𝐺2. 2) Prove 𝜌 is one-to-one (𝜌(𝑎) = 𝜌(𝑏) ⇒ 𝑎 = 𝑏). 3) Prove 𝜌 is onto (for any 𝑏 ∈ 𝐺2, there exists 𝑎 ∈ 𝐺1 such that 𝜌(𝑎) = 𝑏). 4) Prove 𝜌 is operation-preserving (𝜌(𝑎 ∙ 𝑏) = 𝜌(𝑎) ∙ 𝜌(𝑏) for all 𝑎, 𝑏 ∈ 𝐺1).

Theorem 1 (Properties of Isomorphisms)

Let 𝜌: 𝐺1 → 𝐺2 be an isomorphism. Then: 1) 𝜌 carries the identity of 𝐺1 to the identity of 𝐺2. 2) 𝜌 −1: 𝐺2 → 𝐺1 is an isomorphism. 3) 𝜌(𝑎^𝑛) = (𝜌(𝑎))^𝑛 for all 𝑎 ∈ 𝐺1 and for every integer 𝑛. 4) |𝑎| = |𝜌(𝑎)| for all 𝑎 ∈ 𝐺1. 5) |𝐺1| = |𝐺2| in case they are finite groups. 6) For any 𝑎, 𝑏 ∈ 𝐺1, 𝑎 and 𝑏 commute if and only if 𝜌(𝑎) and 𝜌(𝑏) commute. 7) 𝐺1 =

Inner Automorphism (Definition 2)

Let 𝐺 be a group and 𝑎 ∈ 𝐺. The mapping 𝜌𝑎: 𝐺 → 𝐺 defined by 𝜌𝑎(𝑥) = 𝑎𝑥𝑎 −1 is an automorphism, called the inner automorphism of 𝑮 induced by 𝒂.

𝐴𝑢𝑡(𝐺)

The set of all automorphisms of 𝐺.

𝐼𝑛𝑛(𝐺)

The set of all inner automorphisms of 𝐺.

Theorem 2

Let 𝐺 be a group. 1) (𝐴𝑢𝑡(𝐺), ∘) is a group. 2) (𝐼𝑛𝑛(𝐺), ∘) is a group.