Study sheet: Groups, Rings, and Fields

📋 Course Outline

1. Internal Composition Laws
2. Operation Properties and Inverses
3. Groups and Their Examples
4. Group Theorems and Subgroups
5. Group Morphisms
6. Kernels, Images, and Isomorphisms
7. Ring Structure
8. Basic Ring Identities

📖 1. Internal Composition Laws

🔑 Key Concepts & Definitions

• Internal composition law : an application $\varphi:E\times E\to E$ that maps every pair $(a,b)$ to an element denoted $a*b$ in E.

📝 Essential Points

• Addition and multiplication are internal composition laws on ℕ, and intersection and union are internal composition laws on 𝒫(ℕ).

• For functions from E to E, composition is an internal composition law because composing two functions E→E produces another function E→E.

💡 Memory Hook

Internal means the result stays in E; external operations can leave E.

📖 2. Operation Properties and Inverses

🔑 Key Concepts & Definitions

• Commutative law : commutative when $a*b=b*a$ for every pair $(a,b)$ in E².
• Associative law : associative when $(a*b)*c=a*(b*c)$ for every triple $(a,b,c)$ in E³.
• Identity element : an identity element for * when $e*x=x*e=x$ for every x in E.
• Inverse element : If (E,*) has identity e, an element y is an inverse or symmetric of x when $x*y=y*x=e$.

📝 Essential Points

• When a law is associative and commutative, repeated additive operations can be written with Σ and repeated multiplicative operations with Π.

• An element has at most one inverse, and the identity element is its own inverse.

💡 Memory Hook

Closure → associativity → identity → inverse

📖 3. Groups and Their Examples

🔑 Key Concepts & Definitions

• Group : a set G with an internal law * that is associative, has an identity element, and gives every element an inverse.
• Abelian group : A group is called an abelian group when its law is also commutative.

★ Must-know

• (ℕ,+) is not a group, while (ℤ,+) and (ℚ* ,×) are groups.

Further detail

• The unit complex numbers U form an abelian group under multiplication, with identity 1 and inverse of z equal to its conjugate.

• The bijections of a set E form a group under composition, with identity Id_E and inverse f⁻¹ for each bijection f.

💡 Memory Hook

A group has inverses for every element; a monoid-like structure may not.

📖 4. Group Theorems and Subgroups

🔑 Key Concepts & Definitions

• Subgroup : A subset H of a group G is a subgroup when it contains e_G, contains x⁻¹ for every x in H, and is closed under the group law.
• Direct product group : Given groups (G,*) and (H,·), the product G×H is a group under $(x,y)T(x',y')=(x*x',y\cdot y')$, with identity (e_G,e_H) and inverse $(x,y)^{-1}=(x^{-1},y^{-1})$.

📝 Essential Points

• In a group, the identity element is unique and every element has a unique inverse.

📌 For x and y in a group, the inverse of their product is $(x*y)^{-1}=y^{-1}*x^{-1}$, so the order is reversed.

📐 Formula — In a group, the equation $a*x=b$ has the unique solution $x=a^{-1}*b$.

📌 A nonempty subset H of a group G is a subgroup if and only if it contains e and satisfies $x*y^{-1}\in H$ for all x,y in H.

💡 Memory Hook

Identity → inverse → equation → product group → subgroup

📖 5. Group Morphisms

🔑 Key Concepts & Definitions

• Group morphism : A morphism from (G₁,*) to (G₂,

• ) is a function f satisfying $f(x*y)=f(x)\bullet f(y)$ for all x,y in G₁.

★ Must-know

• The three morphism types are:
  • an endomorphism is a morphism from a group to itself
  • an automorphism is a bijective endomorphism
  • an isomorphism is a bijective morphism between two groups

📌 Every group morphism maps the identity to the identity and maps inverses according to $f(x^{-1})=f(x)^{-1}$.

Further detail

📌 The image of a subgroup under a group morphism is a subgroup of the target group, and the inverse image of a subgroup is a subgroup of the source group.

💡 Memory Hook

Endomorphism stays in one group; isomorphism connects two groups bijectively.

📖 6. Kernels, Images, and Isomorphisms

🔑 Key Concepts & Definitions

• Kernel : $\operatorname{Ker}f=\{x\in G₁\mid f(x)=e₂\}$.
• Image : $\operatorname{Im}f=\{y\in G₂\mid \exists x\in G₁, y=f(x)\}$.

★ Must-know

📌 The kernel of a group morphism is a subgroup of the source group and its image is a subgroup of the target group.

📌 A group morphism is injective if and only if its kernel is the trivial subgroup {e₁}.

Further detail

📌 A group morphism is surjective if and only if its image equals the whole target group G₂.

📌 The composition of two group morphisms is a group morphism, and the inverse bijection of a group isomorphism is also a group isomorphism.

💡 Memory Hook

Trivial kernel → injectivity; full image → surjectivity.

📖 7. Ring Structure

🔑 Key Concepts & Definitions

• Ring : a set A equipped with two internal laws + and × such that (A,+) is an abelian group, multiplication is associative and distributive over addition, and multiplication has an identity element 1.

★ Must-know

📐 Formula — Ring multiplication satisfies $x\times(y+z)=x\times y+x\times z$ and $(x+y)\times z=x\times z+y\times z$.

Further detail

• The structures (ℤ,+,×), (ℚ,+,×), (ℝ,+,×), and (ℂ,+,×) are commutative rings.

💡 Memory Hook

Additive group → associative multiplication → distributivity → multiplicative identity

📖 8. Basic Ring Identities

★ Must-know

📌 In any ring, multiplication by zero gives $a\times0=0\times a=0$ for every a.

Further detail

📌 In any ring, multiplication by the additive inverse of 1 gives $(-1)\times a=-a$ for every a.

📌 In any ring, the product of two additive inverses satisfies $(-a)\times(-b)=a\times b$.

💡 Memory Hook

Distributivity over zero forces multiplication by zero to vanish.

📊 Synthesis Tables

Morphisms and Their Properties

Notion	Source and target	Defining property
Endomorphism	Same group	Group morphism
Automorphism	Same group	Bijective endomorphism
Isomorphism	Two groups	Bijective morphism