Algebra
129 questions across 7 topics. Also examined in CNET.
Groups, Subgroups, Sylow Theorems, Cayley's Theorem
- 42Group theory — SL2(Z/2Z)cong S3MSET June 2025Groups, Subgroups, Sylow Theorems, Cayley's TheoremEasy
- 43Group theory — normality from cyclic subgroups; index 2MSET June 2025Groups, Subgroups, Sylow Theorems, Cayley's TheoremMedium
- 44Group theory — the structure of A4MSET June 2025Groups, Subgroups, Sylow Theorems, Cayley's TheoremMedium
- 45Group theory — groups of order p^2: no option always holdsMSET June 2025Groups, Subgroups, Sylow Theorems, Cayley's TheoremMedium
- 48Group theory — conjugacy in S4 is cycle typeMSET June 2025Groups, Subgroups, Sylow Theorems, Cayley's TheoremEasy
- 36Ring theory — four claims, one survivesMSET April 2024Groups, Subgroups, Sylow Theorems, Cayley's TheoremHard
- 37Finite commutative rings — Sylow components as idealsMSET April 2024Groups, Subgroups, Sylow Theorems, Cayley's TheoremHard
- 40Coprime power maps on a finite abelian groupMSET April 2024Groups, Subgroups, Sylow Theorems, Cayley's TheoremMedium
- 42Conjugacy classes — counting them, and what centrality really givesMSET April 2024Groups, Subgroups, Sylow Theorems, Cayley's TheoremHard
- 50The circle is not simply connected; the square disk isMSET April 2024Groups, Subgroups, Sylow Theorems, Cayley's TheoremHard
- 42Which power maps are ring homomorphisms — checking with a concrete counterexampleMSET March 2023Groups, Subgroups, Sylow Theorems, Cayley's TheoremMedium
- 43Finite order of every element does not force a finite groupMSET March 2023Groups, Subgroups, Sylow Theorems, Cayley's TheoremMedium
- 44p-groups always have a non-trivial centerMSET March 2023Groups, Subgroups, Sylow Theorems, Cayley's TheoremHard
- 45An abelian group always has a subgroup of order lcm(m,n)MSET March 2023Groups, Subgroups, Sylow Theorems, Cayley's TheoremMedium
- 46Inn(S3)cong S3, not trivialMSET March 2023Groups, Subgroups, Sylow Theorems, Cayley's TheoremMedium
- 21Equal subgroups characterised via a shared cosetMSET September 2021Groups, Subgroups, Sylow Theorems, Cayley's TheoremHard
- 23Counting elements of each order in S3MSET September 2021Groups, Subgroups, Sylow Theorems, Cayley's TheoremMedium
- 24Counting order-10 elements in a cyclic group of order 20MSET September 2021Groups, Subgroups, Sylow Theorems, Cayley's TheoremEasy
- 53The correct Sylow-counting formulaMSET September 2021Groups, Subgroups, Sylow Theorems, Cayley's TheoremMedium
- 54Counting order-3 elements in mathbf Z9timesmathbf Z3MSET September 2021Groups, Subgroups, Sylow Theorems, Cayley's TheoremEasy
- 62The subgroup lattice of a cyclic group is a product of chainsMSET September 2021Groups, Subgroups, Sylow Theorems, Cayley's TheoremMedium
- 21Homomorphic images of cyclic groups stay cyclic — but the converse failsMSET December 2020Groups, Subgroups, Sylow Theorems, Cayley's TheoremEasy
- 23Cayley's theoremMSET December 2020Groups, Subgroups, Sylow Theorems, Cayley's TheoremEasy
- 53Every prime-power-order group is solvable, even though it need not be abelian or simpleMSET December 2020Groups, Subgroups, Sylow Theorems, Cayley's TheoremMedium
- 55Counting intermediate fields via the subgroup lattice of S3MSET December 2020Groups, Subgroups, Sylow Theorems, Cayley's TheoremEasy
- 21Pairing each element with its inverse forces an odd count of involutionsMSET June 2019Groups, Subgroups, Sylow Theorems, Cayley's TheoremEasy
- 22Every proper-subgroup quotient of (Q,+) is infiniteMSET June 2019Groups, Subgroups, Sylow Theorems, Cayley's TheoremEasy
- 23A prime-order group is cyclic with only the trivial subgroupsMSET June 2019Groups, Subgroups, Sylow Theorems, Cayley's TheoremEasy
- 24Counting GL(2,mathbf Z2) directlyMSET June 2019Groups, Subgroups, Sylow Theorems, Cayley's TheoremEasy
- 53Order 12 is the smallest of the four options admitting non-abelian examplesMSET June 2019Groups, Subgroups, Sylow Theorems, Cayley's TheoremEasy
- 35a^n=e makes the order of a a divisor of nMSET January 2018 Paper IIGroups, Subgroups, Sylow Theorems, Cayley's TheoremEasy
- 36Non-identity self-inverse elements pair up with the rest of G to force parityMSET January 2018 Paper IIGroups, Subgroups, Sylow Theorems, Cayley's TheoremMedium
- 37Groups of order p, p^2 and 15 are abelian; D5 has order 10MSET January 2018 Paper IIGroups, Subgroups, Sylow Theorems, Cayley's TheoremMedium
- 26Groups of order p^2 are abelianMSET January 2018 Paper IIIGroups, Subgroups, Sylow Theorems, Cayley's TheoremEasy
- 27S5 contains the simple non-abelian group A5MSET January 2018 Paper IIIGroups, Subgroups, Sylow Theorems, Cayley's TheoremMedium
- 33Matrices with all entries equal form an abelian groupMSET April 2017 Paper IIGroups, Subgroups, Sylow Theorems, Cayley's TheoremMedium
- 34exp is an isomorphism (R,+)to(R^+,cdot)MSET April 2017 Paper IIGroups, Subgroups, Sylow Theorems, Cayley's TheoremMedium
- 36Aut(Z7)congZ7^, cyclic of order 6MSET April 2017 Paper IIGroups, Subgroups, Sylow Theorems, Cayley's TheoremEasy
- 37Cayley's theorem: G embeds in the symmetric group on its own elementsMSET April 2017 Paper IIGroups, Subgroups, Sylow Theorems, Cayley's TheoremEasy
- 26In a non-abelian group of order p^3, G'=Z(G) has order pMSET April 2017 Paper IIIGroups, Subgroups, Sylow Theorems, Cayley's TheoremHard
- 27A4 has the normal Klein four-groupMSET April 2017 Paper IIIGroups, Subgroups, Sylow Theorems, Cayley's TheoremEasy
- 28Abelian groups of order 3^4 correspond to the 5 partitions of 4MSET April 2017 Paper IIIGroups, Subgroups, Sylow Theorems, Cayley's TheoremEasy
Rings, Ideals, UFD, PID, Euclidean Domain
- 40Algebra — subrings of UFDs and gcdsMSET June 2025Rings, Ideals, UFD, PID, Euclidean DomainMedium
- 41Algebra — small non-commutative ringsMSET June 2025Rings, Ideals, UFD, PID, Euclidean DomainMedium
- 38Irreducible but not prime — the classic mathbf Z[sqrt-5] witnessMSET April 2024Rings, Ideals, UFD, PID, Euclidean DomainHard
- 39Prime vs. maximal ideals — where the general claim breaksMSET April 2024Rings, Ideals, UFD, PID, Euclidean DomainHard
- 39Maximal principal ideals of F[x] are generated by irreduciblesMSET March 2023Rings, Ideals, UFD, PID, Euclidean DomainEasy
- 40Spotting the false claim about Euclidean domains and PIDsMSET March 2023Rings, Ideals, UFD, PID, Euclidean DomainMedium
- 41mathbf Z[x] is neither a PID nor EuclideanMSET March 2023Rings, Ideals, UFD, PID, Euclidean DomainEasy
- 22An ideal generated by a unit is the whole ringMSET September 2021Rings, Ideals, UFD, PID, Euclidean DomainMedium
- 25A domain that is not a field — the polynomial ring over mathbf RMSET September 2021Rings, Ideals, UFD, PID, Euclidean DomainEasy
- 22X^2-n is irreducible for squarefree n1, giving a field — hence, in particular, a domainMSET December 2020Rings, Ideals, UFD, PID, Euclidean DomainEasy
- 25Recognising when a two-generator ideal collapses to a principal oneMSET December 2020Rings, Ideals, UFD, PID, Euclidean DomainMedium
- 48Z[sqrt-p] fails unique factorization for every odd prime pMSET December 2020Rings, Ideals, UFD, PID, Euclidean DomainMedium
- 62A lattice's ideals are all prime exactly when the lattice is totally orderedMSET December 2020Rings, Ideals, UFD, PID, Euclidean DomainEasy
- 25A one-variable polynomial ring over a field is always a PIDMSET June 2019Rings, Ideals, UFD, PID, Euclidean DomainEasy
- 48The classic irreducible-but-not-prime element on the coordinate ring of the circleMSET June 2019Rings, Ideals, UFD, PID, Euclidean DomainEasy
- 33Polynomials with zero constant term form a ring without 1MSET January 2018 Paper IIRings, Ideals, UFD, PID, Euclidean DomainEasy
- 10A finite Boolean ring is a power of Z2MSET January 2018 Paper IIIRings, Ideals, UFD, PID, Euclidean DomainMedium
- 11In a finite ring the additive order of 1 is finiteMSET January 2018 Paper IIIRings, Ideals, UFD, PID, Euclidean DomainMedium
- 13Z[x] is a UFD in which (2,x) is not principalMSET January 2018 Paper IIIRings, Ideals, UFD, PID, Euclidean DomainEasy
- 14Q[sqrt2] sits inside RsubsetCMSET January 2018 Paper IIIRings, Ideals, UFD, PID, Euclidean DomainMedium
- 28Infinitely many variables give a strictly increasing chain of idealsMSET January 2018 Paper IIIRings, Ideals, UFD, PID, Euclidean DomainEasy
- 35If A kills no non-zero matrix from the left, A has trivial kernelMSET April 2017 Paper IIRings, Ideals, UFD, PID, Euclidean DomainMedium
- 116=2cdot3=(1+sqrt-5)(1-sqrt-5) in Z[sqrt-5]MSET April 2017 Paper IIIRings, Ideals, UFD, PID, Euclidean DomainEasy
- 12Real quadratic rings have infinitely many units: (1+sqrt2)^nMSET April 2017 Paper IIIRings, Ideals, UFD, PID, Euclidean DomainEasy
Permutations, Combinations, Pigeon-Hole Principle, Inclusion-Exclusion
- 10Counting subsetsMSET June 2025Permutations, Combinations, Pigeon-Hole Principle, Inclusion-ExclusionEasy
- 2Combinatorics — subsets built from even elementsMSET April 2024Permutations, Combinations, Pigeon-Hole Principle, Inclusion-ExclusionEasy
- 41The inversion map — a homomorphism exactly on abelian groupsMSET April 2024Permutations, Combinations, Pigeon-Hole Principle, Inclusion-ExclusionHard
- 43Conjugacy in An — when Sn-classes splitMSET April 2024Permutations, Combinations, Pigeon-Hole Principle, Inclusion-ExclusionHard
- 45Counting arrangements of distinct lettersMSET April 2024Permutations, Combinations, Pigeon-Hole Principle, Inclusion-ExclusionEasy
- 46Pigeonhole — forcing a shared day of the weekMSET April 2024Permutations, Combinations, Pigeon-Hole Principle, Inclusion-ExclusionMedium
- 104Inclusion–exclusion — minimum possible joint attendanceMSET April 2024Permutations, Combinations, Pigeon-Hole Principle, Inclusion-ExclusionHard
- 47Counting distinct monomial terms in a multinomial expansionMSET March 2023Permutations, Combinations, Pigeon-Hole Principle, Inclusion-ExclusionEasy
- 104A simple counting-principle questionMSET March 2023Permutations, Combinations, Pigeon-Hole Principle, Inclusion-ExclusionEasy
- 64A coefficient forced to vanish by a minimum-degree argumentMSET September 2021Permutations, Combinations, Pigeon-Hole Principle, Inclusion-ExclusionEasy
- 68Binary representation — sums of distinct powers of 2MSET September 2021Permutations, Combinations, Pigeon-Hole Principle, Inclusion-ExclusionEasy
- 24The order of a permutation is the lcm of its disjoint cycle lengthsMSET December 2020Permutations, Combinations, Pigeon-Hole Principle, Inclusion-ExclusionMedium
- 64Pigeonhole: one more than the number of couples forces a full pairMSET December 2020Permutations, Combinations, Pigeon-Hole Principle, Inclusion-ExclusionEasy
- 96Inclusion–exclusion for integers avoiding three small prime divisorsMSET December 2020Permutations, Combinations, Pigeon-Hole Principle, Inclusion-ExclusionEasy
- 64Counting alternating arrangements out of all permutationsMSET June 2019Permutations, Combinations, Pigeon-Hole Principle, Inclusion-ExclusionEasy
- 40Multinomial coefficient 7!/2!,2!,3!MSET January 2018 Paper IIIPermutations, Combinations, Pigeon-Hole Principle, Inclusion-ExclusionEasy
- 42Inclusion–exclusion with the lcms 30, 40, 24 and 120MSET January 2018 Paper IIIPermutations, Combinations, Pigeon-Hole Principle, Inclusion-ExclusionEasy
- 40Stars and bars: binom10+6-16-1=binom155MSET April 2017 Paper IIIPermutations, Combinations, Pigeon-Hole Principle, Inclusion-ExclusionEasy
Fundamental Theorem of Arithmetic, Congruences, Chinese Remainder Theorem
- 46Number theory — the Frobenius number of 3 and 8MSET June 2025Fundamental Theorem of Arithmetic, Congruences, Chinese Remainder TheoremMedium
- 47Number theory — units of Z12MSET June 2025Fundamental Theorem of Arithmetic, Congruences, Chinese Remainder TheoremEasy
- 48Last digit of 3^55 via the period-4 cycleMSET March 2023Fundamental Theorem of Arithmetic, Congruences, Chinese Remainder TheoremEasy
- 69Solving a system of three congruences by direct verificationMSET September 2021Fundamental Theorem of Arithmetic, Congruences, Chinese Remainder TheoremEasy
- 70Using Wilson's theorem to evaluate (50!)^2bmod101MSET September 2021Fundamental Theorem of Arithmetic, Congruences, Chinese Remainder TheoremEasy
- 68Infinitely many integer solutions of 7x-8y=5, spaced 8 apart in xMSET December 2020Fundamental Theorem of Arithmetic, Congruences, Chinese Remainder TheoremMedium
- 54Z72congZ9timesZ8 by the Chinese Remainder TheoremMSET June 2019Fundamental Theorem of Arithmetic, Congruences, Chinese Remainder TheoremEasy
- 68Squares mod 3 are exactly 0,1MSET June 2019Fundamental Theorem of Arithmetic, Congruences, Chinese Remainder TheoremEasy
- 69Computing 3^123bmod100 by repeated squaringMSET June 2019Fundamental Theorem of Arithmetic, Congruences, Chinese Remainder TheoremEasy
- 70Fermat's two-square theorem flags the prime 11equiv3pmod4 to an odd powerMSET June 2019Fundamental Theorem of Arithmetic, Congruences, Chinese Remainder TheoremEasy
- 49Legendre's three-square theorem excludes 4^a(8b+7)MSET January 2018 Paper IIIFundamental Theorem of Arithmetic, Congruences, Chinese Remainder TheoremMedium
- 502 is a square mod p iff pequivpm1pmod8MSET January 2018 Paper IIIFundamental Theorem of Arithmetic, Congruences, Chinese Remainder TheoremMedium
- 51Gauss's identity sumdmid nvarphi(d)=nMSET January 2018 Paper IIIFundamental Theorem of Arithmetic, Congruences, Chinese Remainder TheoremEasy
- 43For odd m, 2^varphi(m)equiv1pmod mMSET April 2017 Paper IIIFundamental Theorem of Arithmetic, Congruences, Chinese Remainder TheoremMedium
- 49ax+bequiv0pmod m is solvable iff gcd(a,m)mid bMSET April 2017 Paper IIIFundamental Theorem of Arithmetic, Congruences, Chinese Remainder TheoremEasy
- 50Primes equiv3pmod4 must occur to even powers: 180=2^2cdot3^2cdot5MSET April 2017 Paper IIIFundamental Theorem of Arithmetic, Congruences, Chinese Remainder TheoremEasy
- 51sumdmid nmu(d)=0 for n1MSET April 2017 Paper IIIFundamental Theorem of Arithmetic, Congruences, Chinese Remainder TheoremEasy
Fields, Finite Fields, Field Extensions, Galois Theory
- 36Algebra — Galois group of an irreducible cubicMSET June 2025Fields, Finite Fields, Field Extensions, Galois TheoryMedium
- 37Algebra — subfields of mathbb F64MSET June 2025Fields, Finite Fields, Field Extensions, Galois TheoryEasy
- 38Algebra — finite fields are never algebraically closedMSET June 2025Fields, Finite Fields, Field Extensions, Galois TheoryEasy
- 47Finite fields — the sum and product of all elementsMSET April 2024Fields, Finite Fields, Field Extensions, Galois TheoryHard
- 48Finite fields — the canonical splitting-field descriptionMSET April 2024Fields, Finite Fields, Field Extensions, Galois TheoryHard
- 37Spotting the false claim among four algebra factsMSET March 2023Fields, Finite Fields, Field Extensions, Galois TheoryMedium
- 38Counting low-degree irreducible polynomials over mathbf Z2MSET March 2023Fields, Finite Fields, Field Extensions, Galois TheoryEasy
- 47Endomorphisms of an algebraic extension are automatically bijectiveMSET September 2021Fields, Finite Fields, Field Extensions, Galois TheoryHard
- 55Comparing degree-8 and degree-2 field extensionsMSET September 2021Fields, Finite Fields, Field Extensions, Galois TheoryMedium
- 47The unit group of mathbf Fp is simple exactly when p-1 is primeMSET June 2019Fields, Finite Fields, Field Extensions, Galois TheoryMedium
- 55Finite field extensions are always cyclic Galois, via the FrobeniusMSET June 2019Fields, Finite Fields, Field Extensions, Galois TheoryMedium
- 38Subfields of mathbb F2^7 correspond to divisors of the prime 7MSET January 2018 Paper IIFields, Finite Fields, Field Extensions, Galois TheoryMedium
- 12mathbb Fp(x) is an infinite field containing ZpMSET January 2018 Paper IIIFields, Finite Fields, Field Extensions, Galois TheoryMedium
- 29Q(sqrt[3]2,omega) has degree 3cdot2=6MSET January 2018 Paper IIIFields, Finite Fields, Field Extensions, Galois TheoryMedium
- 30A regular heptagon is not constructible: 7 is not a Fermat primeMSET January 2018 Paper IIIFields, Finite Fields, Field Extensions, Galois TheoryMedium
- 38The multiplicative group of a finite field is cyclicMSET April 2017 Paper IIFields, Finite Fields, Field Extensions, Galois TheoryEasy
- 30sqrt[3]2 has degree 3, not a power of 2MSET April 2017 Paper IIIFields, Finite Fields, Field Extensions, Galois TheoryEasy
Polynomial Rings and Irreducibility Criteria
- 35Algebra — minimal polynomial of sqrt2+sqrt3MSET June 2025Polynomial Rings and Irreducibility CriteriaEasy
- 39Algebra — irreducibility of x^3-9 modulo nMSET June 2025Polynomial Rings and Irreducibility CriteriaMedium
- 35Eisenstein's criterion — and its limitsMSET April 2024Polynomial Rings and Irreducibility CriteriaMedium
- 48X^16+1 has no real roots — hence no linear factorsMSET September 2021Polynomial Rings and Irreducibility CriteriaEasy
- 13mathbb Fp^n exists for every n, so irreducibles of every degree existMSET April 2017 Paper IIIPolynomial Rings and Irreducibility CriteriaMedium
- 291/(1+alpha)inQ[alpha] exactly when alpha is algebraicMSET April 2017 Paper IIIPolynomial Rings and Irreducibility CriteriaMedium
Lattice Theory (Distributive/Modular/Boolean Lattices)
- 62Completeness gives boundedness, and Boolean lattices are always distributiveMSET June 2019Lattice Theory (Distributive/Modular/Boolean Lattices)Medium
- 96Excluding the pentagon N5 from the five lattices on 5 elementsMSET June 2019Lattice Theory (Distributive/Modular/Boolean Lattices)Easy
- 97Neither distributivity nor modularity alone guarantees complementation existsMSET June 2019Lattice Theory (Distributive/Modular/Boolean Lattices)Medium
- 41The dual swaps wedge and vee after removing the implicationMSET January 2018 Paper IIILattice Theory (Distributive/Modular/Boolean Lattices)Easy
- 41The divisor lattice is graded: every maximal chain has the same lengthMSET April 2017 Paper IIILattice Theory (Distributive/Modular/Boolean Lattices)Medium