Abstract Algebra - Groups & Rings

Q1: Let G be a group of order 2023. Which of the following statements is necessarily TRUE?

G has a subgroup of order 17 G has a subgroup of order 7 G has a subgroup of order 119 G is cyclic

Q2: Let R be a commutative ring with unity. If every ideal of R is prime, then which of the following is TRUE?

R is a field R is an integral domain R is a PID R has exactly one prime ideal

Q3: Let G be a finite group and H be a subgroup of index 2. Which of the following is ALWAYS TRUE?

H is normal in G H is abelian G/H is cyclic H is a maximal subgroup

Q4: In a ring R, an element a is called nilpotent if aⁿ = 0 for some positive integer n. If R is a commutative ring with unity and a, b are nilpotent, which of the following is TRUE?

a + b is nilpotent a + b is always a unit ab is nilpotent only if a = 0 or b = 0 a + b is never nilpotent

Q5: Let G be a group of order 105. Which of the following is NOT necessarily true?

G has a normal Sylow 5-subgroup G has a normal Sylow 7-subgroup G has a normal Sylow 3-subgroup G is solvable

Q6: Which of the following rings is NOT an integral domain?

Z[i] (Gaussian integers) Z[√2] Z/6Z Z[x] (polynomials over integers)

Q7: Let G be a group where every element is its own inverse (i.e., g² = e for all g ∈ G). Which of the following is TRUE?

G is abelian G is cyclic G is finite G is a simple group

Q8: In a principal ideal domain (PID), which of the following statements is FALSE?

Every ideal is finitely generated Every prime ideal is maximal Euclidean domains are PIDs PIDs are integral domains

Q9: Let G be a group of order p² where p is prime. Which of the following is TRUE?

G is always abelian G is always cyclic G is always simple G is always non-abelian

Q10: In the ring of integers Z, which of the following ideals is NOT prime?

(2) (3) (5) (4)

Q11: Let G be a finite abelian group. Which of the following statements is TRUE?

G is isomorphic to a direct product of cyclic groups of prime power order G has no subgroups of composite order G is always cyclic G has a unique subgroup of each order dividing |G|

Q12: For a ring R with unity, the Jacobson radical J(R) is defined as the intersection of all maximal left ideals. Which of the following is TRUE?

J(R) is a two-sided ideal J(R) contains all nilpotent elements J(R) = {0} iff R is a field J(R) is always principal

Q13: Let G be a group and N be a normal subgroup of G. Which of the following is ALWAYS TRUE about the quotient group G/N?

If G is abelian, then G/N is abelian If G/N is cyclic, then G is cyclic If G/N is abelian, then G is abelian If N is finite, then G is finite

Q14: In the ring of Gaussian integers Z[i], which of the following is a prime element?

2 3 1 + i 5

Q15: Let G be a group of order 36. Which of the following is TRUE about the Sylow 3-subgroups?

The number of Sylow 3-subgroups is 1 The number of Sylow 3-subgroups is 4 The number of Sylow 3-subgroups divides 4 and is ≡ 1 mod 3 There is no Sylow 3-subgroup

Q16: Which of the following is a Euclidean domain?

Z[x] (polynomials over integers) Z[√−5] Z/6Z Z[i] (Gaussian integers)