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The Problem Bank (List)

Below is the current master list of all problems in the Problem Bank. For a quick view of any problem, simply hover your mouse over the link.

Problem Problem Type
A condition ensuring diagonalizability
A condition to be non-cyclic
A condition under which a group must be abelian
A condition under which a group must be abelian (2)
A dihedral group that is not an internal direct product
A group isomorphic to a subgroup of a direct product of quotient groups
A group isomorphic to an internal direct product
A group of upper-triangular matrices
A group with a trivial automorphism group
A group with a trivial automorphism group (2)
A linear transformation from a vector space of polynomials
A maximal ideal in a function ring
A non-PID
A property of surjective linear transformations
A property of the order of an element
A ring in which all prime ideals are maximal
A vector space of square matrices
An automorphism of a group of odd order
An evaluation morphism
An isomorphism of rings
An upper bound on the number of nonzero eigenvalues
Analyzing an unusual matrix
Another condition for a group to be abelian
Automorphisms of a finite cyclic group
Automorphisms of a ring
Bases for a subspace and its orthogonal complement
Boolean algebras
Boolean rings are commutative
Centralizers in symmetric groups
Characteristic of a ring
Characteristic of a ring (2)
Closely related subgroups of a finite group
Comparing cosets
Computations in symmetric groups
Computations with a given linear transformation
Computations with inner automorphisms
Computing an automorphism group
Constructing a field extension
Constructing the field with eight elements
Counting morphisms between specified groups
Diagonalization and matrix powers
Diagonalization of a given matrix
Diagonalization of a given matrix (2)
Dimension of a PID
Dimension of a subspace and its orthogonal complement
Direct sums and idempotent transformations
Directly proving the existence of an element of a desired order
Dot product and cross product as linear transformations
Eigenvalues and eigenspaces of a matrix with a given property
Eigenvectors and linear independence
Eigenvectors of commuting linear transformations
Eigenvectors with distinct eigenvalues are linearly independent
Elements of finite order
Elements of order 2
Euclidean domains are PIDs
Evaluation at i
Existence of a normal subgroup of finite index
Existence of an identity element in a finite ring
Existence of an identity element in a group
Existence of automorphisms
Existence of certain ring morphisms
Finding all morphisms between two groups
Generator for a field extension
Group of units of a product
Ideals in a polynomial ring
Ideals in a polynomial ring (2)
Idempotent elements in a ring
Idempotent elements in a ring (2)
Image of a normal subgroup and induced morphisms
Image of an evaluation morphism
Image of the identity element under a ring morphism
Image of the identity is the identity
Indices and intersections
Inner and outer automorphisms
Inner automorphisms and the center of a group
Inner automorphisms of an alternating group
Jordan canonical form of a matrix
Jordan canonical form of a matrix (2)
Linear dependence and relations
Linear endomorphism of a vector space of matrices
Linear transformation from a vector space of polynomials
Matrix and characteristic polynomial for a given linear transformation
Matrix and eigenvalues of a given linear transformation
Matrix representing a linear transformation
Maximal ideals in a PID
Morphism from the Gaussian integers
Morphism from the Gaussian integers (2)
Nilpotent elements
Nilpotent elements in a ring
Nonexistence of a simple group of a given order
Nonexistence of morphisms between two groups
Nonexistence of small nonabelian groups
Nonzero prime ideals are maximal in a PID
Normal subgroups with trivial intersection
Normality and the operation on cosets
Normality and the operation on cosets (defunct)
Normalizer of a subgroup
Normalizers and centralizers
Numerical range of a linear transformation
Order of a power of an element
Order of an element in a finite group
Order of elements in a symmetric group
Orders of elements in a quotient group
Orthogonal complements
Orthogonal projection and scaling
Orthogonal projection onto a line
Orthogonal projection onto a line (2)
Orthogonal projection onto a line (3)
Orthogonal projection onto a plane
Orthogonal projection onto a plane (2)
Polynomials with even constant term
Preimage of a subgroup
Prime and irreducible elements in a commutative ring
Prime ideals and quotient rings
Product of two subgroups
Products of quotient groups
Projection onto a plane
Projection onto a quotient
Projections and adjoints
Properties of Boolean rings
Properties of the annihilator
Properties of the center of a group
Properties of transpose
Property of the order of an element
Proving an ideal is prime
Proving Lagrange's Theorem
Quotienting out nilpotent elements
Quotients and direct products
Radial expansion from a fixed line
Reflection across a plane
Rotation around an axis
Skew-symmetric matrices
Stabilizer of a coset
Subgroups of a group of even order
Sum and union of subspaces
The cyclic group of order 2020
The field with eight elements
The field with nine elements
The integers as a subgroup of the rationals
The kernel of an evaluation morphism
The nilradical of a ring
The structure of the integers as both a group and a ring
The Third Isomorphism Theorem
Using the Chinese Remainder Theorem
Verifying axioms of a group
Working modulo 11