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Quizzes > Mathematics & Statistics

Abstract Algebra I Quiz

Free Practice Quiz & Exam Preparation

Difficulty: Moderate
Questions: 15
Study OutcomesAdditional Reading
3D voxel art representing the course Abstract Algebra I

This engaging practice quiz for Abstract Algebra I is designed to reinforce your understanding of key topics including isomorphism theorems for groups, group actions, and composition series, along with an in-depth exploration of field extensions and the fundamentals of Galois theory. Additionally, you'll tackle challenging concepts such as solvable and nilpotent groups, modules over commutative rings, and matrix canonical forms, making it an ideal resource for students aiming to excel in both theory and application.

Which statement correctly represents the First Isomorphism Theorem for groups?
If φ: G → H is a group homomorphism, then G/ker(φ) ≅ im(φ).
Every group homomorphism is injective, so G is isomorphic to H.
The kernel of any homomorphism is trivial, leading to an isomorphism between G and im(φ).
Every quotient group is isomorphic to its corresponding subgroup in H.
The First Isomorphism Theorem states that for a group homomorphism, the quotient of the domain by the kernel is isomorphic to the image. This fundamental theorem simplifies the study of group homomorphisms by connecting kernels and images.
Which of the following definitions best describes a group action on a set X?
A group G acts on a set X if there is a map G × X → X satisfying e.x = x and (gh).x = g.(h.x) for all g, h in G and x in X.
A group action is any mapping from X to G that is one-to-one.
A group action only requires that each element of G fixes at least one element of X.
A group action is defined only when G is abelian and X is a subgroup of G.
A group action is defined by a function from G × X to X that satisfies two axioms: the identity of the group acts as the identity and the action respects the group operation. These conditions ensure that the structure of the group meaningfully influences the set X.
Which condition is necessary for a subgroup chain to qualify as a composition series?
Each consecutive factor group is simple.
Each factor group is cyclic.
The series must consist of normal subgroups only.
The chain must have infinitely many subgroups.
A composition series is defined as a finite sequence of subgroups where each successive factor group is simple. This maximal refinement of subgroups into simple groups is central to the Jordan-Hölder theorem.
What does the Jordan-Hölder theorem guarantee about composition series of a finite group?
Any two composition series have the same length and the same factors up to isomorphism and order.
Every subgroup in the series is normal in the entire group.
All composition series are identical in the sequence of subgroups.
Each composition series only consists of cyclic factor groups.
The Jordan-Hölder theorem asserts that although different composition series might list subgroups in a different order, the factors (simple groups) are uniquely determined up to permutation and isomorphism. This uniqueness provides an invariant that characterizes the structure of the group.
Which of the following properties defines a nilpotent group?
It has an upper central series that terminates with the group itself.
Every subgroup of the group is normal.
The group has no nontrivial proper subgroups.
The group is abelian, and all its elements commute.
A nilpotent group is characterized by the existence of an upper central series that eventually reaches the entire group. This series highlights a structured form of commutativity even in groups that might not be abelian.
Which of the following is true for a solvable group?
It has a subnormal series whose successive quotients are abelian.
It is always cyclic and generated by a single element.
All its subgroups are simple and non-abelian.
Its commutator subgroup is trivial.
A group is solvable if there exists a chain of subgroups such that each quotient is abelian. This property is central in group theory and has deep implications in areas like Galois theory.
Let F be a field and E a field extension of F. Which statement accurately describes an algebraic extension?
Every element in E satisfies a nonzero polynomial with coefficients in F.
There exists at least one element in E that does not satisfy any polynomial over F.
E is generated by a transcendental element over F.
The extension is always finite in degree.
An extension is algebraic when every element of the extended field is a root of some nonzero polynomial with coefficients in the base field F. This distinction is key in differentiating algebraic extensions from transcendental ones.
What distinguishes a transcendental extension from an algebraic extension?
It contains at least one element that is not a root of any nonzero polynomial with coefficients in the base field.
All elements in the extension are roots of linear polynomials over the base field.
The extension is always finite, unlike algebraic extensions.
Every element in the extension satisfies a unique polynomial over the base field.
In a transcendental extension, at least one element does not satisfy any polynomial equation with coefficients in F, distinguishing it from an algebraic extension where all elements do. This property is essential in classifying different types of field extensions.
Which statement best describes an algebraically closed field?
Every nonconstant polynomial with coefficients in the field has a root within the field.
Every polynomial splits into linear factors only after extending to a larger field.
The field has no nontrivial field extensions.
All elements of the field are algebraic over the rational numbers.
An algebraically closed field is one in which every nonconstant polynomial has a root within the field itself. This is a crucial property that underlies many results in field theory and complex analysis.
In Galois theory, what does the Fundamental Theorem of Galois Theory establish?
A bijective correspondence between the subgroups of a Galois group and the intermediate fields of the extension.
That every intermediate field of an extension is a normal extension of the base field.
That every subgroup of the Galois group must be cyclic.
That the degree of the extension equals the number of intermediate fields.
The Fundamental Theorem of Galois Theory creates a one-to-one correspondence between the subgroups of the Galois group and the intermediate fields associated with the field extension. This correspondence is essential for understanding the symmetry and structure of field extensions.
For a finitely generated module over a principal ideal domain, which statement correctly describes its structure?
It is isomorphic to a direct sum of cyclic modules, including both free and torsion parts.
It is always a free module with no torsion.
It decomposes into an infinite direct sum of simple modules.
It cannot have a direct sum decomposition over a PID.
The structure theorem for finitely generated modules over a PID states that such modules can be decomposed as a direct sum of cyclic modules, partitioning into free parts and torsion parts. This result is fundamental in both module theory and its applications to finite abelian groups.
Which of the following is a direct application of the structure theorem for finitely generated modules over a PID?
Classifying finite abelian groups as direct sums of cyclic groups.
Determining the solvability of a group via its subgroup structure.
Proving that every matrix is similar to a diagonal matrix.
Establishing that every field extension is simple.
The structure theorem is pivotal in classifying finite abelian groups, as it shows that these groups can be expressed as direct sums of cyclic groups. This classification has far-reaching consequences in understanding the underlying structure of abelian groups.
What is a key characteristic of the elementary divisors form used in matrix canonical forms?
It expresses the module as a direct sum of primary cyclic submodules.
It guarantees that the matrix is diagonalizable over any field.
It can only be applied to matrices with distinct eigenvalues.
It represents the matrix exactly in Jordan canonical form.
The elementary divisors form breaks a module into cyclic parts that are associated with prime power factors. This decomposition is critical in understanding the module's structure, particularly in the context of matrix canonical forms over a principal ideal domain.
Consider a group action of G on a set X. What does the orbit-stabilizer theorem state?
The size of the orbit of an element multiplied by the size of its stabilizer equals the order of the group.
The size of the orbit is always equal to the order of the subgroup acting on X.
Every group action satisfies that all orbits have the same size.
The orbit of every element is the entire set X.
The orbit-stabilizer theorem provides a fundamental relation in group actions: the product of the size of an element's orbit and the size of its stabilizer equals the order of the group. This theorem is an essential tool for counting and classifying orbits in group actions.
In the study of field extensions, which statement correctly describes the degree of an extension?
It is defined as the dimension of the extension field considered as a vector space over the base field.
It is the total number of elements present in the extension field.
It always represents an infinite measure whenever the extension is transcendental.
It is calculated by summing the degrees of the minimal polynomials of each element in the extension.
The degree of a field extension is quantified by the dimension of the extension field as a vector space over the base field F. This concept is fundamental in characterizing the size and structure of field extensions, particularly in Galois theory.
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Study Outcomes

  1. Analyze group structures using isomorphism theorems and composition series.
  2. Apply group action concepts to determine orbit and stabilizer properties.
  3. Evaluate field extensions, including algebraic and transcendental cases, and use Galois theory to solve related problems.
  4. Examine the structure of finitely generated modules over principal ideal domains and apply this understanding to finite Abelian groups and matrix canonical forms.

Abstract Algebra I Additional Reading

Embarking on your Abstract Algebra journey? Here are some top-notch resources to guide you through the twists and turns of group theory, field extensions, and more:

  1. Abstract Algebra 1 - Spring 2025 | Ýlvaro Lozano-Robledo Dive into the fundamental topics of modern algebra with Professor Lozano-Robledo's course materials, including lecture notes and video resources.
  2. Abstract Algebra Online | Department of Mathematical Sciences | Northern Illinois University Explore a comprehensive collection of definitions, theorems, and solved problems tailored for undergraduate and first-year graduate students.
  3. Math 500, Abstract Algebra I Access detailed lecture notes, homework assignments, and exams from Professor Nathan Dunfield's graduate course at the University of Illinois.
  4. Galois Theory - a first course Delve into Brent Everitt's self-contained introduction to Galois theory, perfect for students with a foundational understanding of abstract algebra.
  5. Abstract Algebra I (Rutgers 16:640:551, Fall 2019) Peruse lecture notes from Rutgers University's graduate course, covering topics from group theory to modules over principal ideal domains.
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