Flashcards: Groups, Rings, and Fields — 48 cards

What is an internal composition law on a set E?

It is an application $\varphi:E\times E\to E$ mapping pairs to elements in E.

Which operations are internal composition laws on ℕ?

Addition and multiplication are internal composition laws on ℕ.

Which operations are internal composition laws on 𝒫(ℕ)?

Intersection and union are internal composition laws on 𝒫(ℕ).

Why is composition of functions from E to E an internal composition law?

Because composing two functions E→E produces another function E→E.

When is an internal law * commutative?

When $a*b=b*a$ for every pair $(a,b)$ in $E^2$.

When is an internal law * associative?

When $(a*b)*c=a*(b*c)$ for every triple $(a,b,c)$ in $E^3$.

What defines an identity element e for *?

For every $x$ in $E$, $e*x=x*e=x$ holds.

What is an inverse element y of x in (E,*) with identity e?

When $x*y=y*x=e$ holds.

What notation is used for repeated additive and multiplicative operations if the law is associative and commutative?

Repeated additive operations use $\Sigma$ and multiplicative operations use $\Pi$.

How many inverses can an element have?

An element has at most one inverse.

What is special about the identity element's inverse?

The identity element is its own inverse.

What are the three main properties of a group law * on set G?

Associativity, identity element, and inverses for every element.

When is a group called an abelian group?

When its law is commutative.

Which of (ℕ,+), (ℤ,+), and (ℚ*,×) are groups?

(ℤ,+) and (ℚ*,×) are groups, but (ℕ,+) is not.

What identity element do unit complex numbers U have under multiplication?

The identity element is 1.

What is the inverse of a unit complex number z in U under multiplication?

Its conjugate is the inverse of z.

What operation forms a group on the bijections of a set E?

Composition of functions forms the group operation.

What is the identity element in the group of bijections of E?

The identity function Id_E is the identity element.

What is the inverse of a bijection f in the group of bijections of E?

The inverse function f⁻¹ is the inverse of f.

Is the identity element in a group unique?

Yes, the identity element in a group is unique.

Does every element in a group have a unique inverse?

Every element in a group has a unique inverse.

What is the inverse of the product $x*y$ in a group?

The inverse is $(x*y)^{-1} = y^{-1} * x^{-1}$.

How do you solve the equation $a*x = b$ in a group?

The unique solution is $x = a^{-1} * b$.

What is the identity element in the direct product group $G \times H$?

The identity is $(e_G, e_H)$.

How is the inverse of an element $(x,y)$ defined in $G \times H$?

The inverse is $(x,y)^{-1} = (x^{-1}, y^{-1})$.

What conditions define a subgroup H of a group G?

H contains $e_G$, inverses of its elements, and is closed under the group law.

What is the subgroup test for a nonempty subset H of G?

H is a subgroup if $x*y^{-1} \in H$ for all $x,y \in H$ and H contains $e$.

What condition defines a group morphism f from (G₁,*) to (G₂,•)?

It satisfies $f(x*y)=f(x)\bullet f(y)$ for all x,y in G₁.

What is an endomorphism in group theory?

A morphism from a group to itself.

What distinguishes an automorphism from an endomorphism?

An automorphism is a bijective endomorphism.

What defines an isomorphism between two groups?

A bijective morphism between two groups.

What does every group morphism do to the identity element?

It maps the identity to the identity.

How does a group morphism map inverses?

It satisfies $f(x^{-1})=f(x)^{-1}$.

What is the image of a subgroup under a group morphism?

A subgroup of the target group.

What is the inverse image of a subgroup under a group morphism?

A subgroup of the source group.

What is the kernel of a group morphism f:G₁→G₂?

The set of elements in G₁ mapped to the identity e₂ in G₂.

What is the image of a group morphism f:G₁→G₂?

The set of elements in G₂ that are images of elements from G₁.

What subgroup properties do the kernel and image of a group morphism have?

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

When is a group morphism injective in terms of its kernel?

If and only if its kernel is the trivial subgroup {e₁}.

When is a group morphism surjective regarding its image?

If and only if its image equals the entire target group G₂.

What is true about the composition of two group morphisms?

Their composition is also a group morphism.

What is true about the inverse of a group isomorphism?

Its inverse bijection is also a group isomorphism.

What is a ring in algebra?

A set with two internal laws + and × where (A,+) is an abelian group, multiplication is associative, distributive, and has identity 1.

What distributive laws does ring multiplication satisfy?

It satisfies left and right distributivity over addition.

Which number sets form commutative rings with + and ×?

The integers, rationals, reals, and complex numbers.

What is the result of multiplying any element by zero in a ring?

The result is zero.

What does multiplying any element by -1 yield in a ring?

It yields the additive inverse of that element.

What is the product of two additive inverses in a ring?

It equals the product of the original elements.