Abelian Groups - Definition

An abelian group is a set 𝐺 with one binary operation on 𝐺 satisfying the abelian group axioms (𝑎, 𝑏, 𝑐 are arbitrary elements of the abelian group 𝐺)

A binary operation on 𝐺 is a mapping 𝐺×𝐺 → 𝐺, that is, a correspondence that associates with each ordered pair of elements of 𝐺 a uniquely determined element of 𝐺.

	Binary Operation Properties
	Closed	Associativity	Identity	Invertibility	Commutativity
Binary Operation	REQUIRED (𝑎 · 𝑏) ∊ 𝐹 (𝑎 + 𝑏) ∊ 𝐹	REQUIRED 𝑎 · (𝑏 · 𝑐) = (𝑎 · 𝑏) · 𝑐 𝑎 + (𝑏 + 𝑐) = (𝑎 + 𝑏) + 𝑐	REQUIRED 𝑎 · 1 = 𝑎 𝑎 + 0 = 𝑎	REQUIRED 𝑎 · 𝑎-1 = 1 𝑎 + (-𝑎) = 0	REQUIRED 𝑎 · 𝑏 = 𝑏 · 𝑎 𝑎 + 𝑏 = 𝑏 + 𝑎

／var／log marcus chiu

Explorer

Abelian Groups (Algebraic Structure)

Abelian Groups (Algebraic Structure)

Abelian Groups - Definition

／var／logmarcus chiu

Explorer

Abelian Groups (Algebraic Structure)

Abelian Groups (Algebraic Structure)

Abelian Groups - Definition

／var／log marcus chiu