BMath Bachelor of Mathematics Abstract Algebra and Group Theory 1 — Questions and Answers
Question 1: Which of the following is NOT a required property for a set G with binary operation * to be a group?
- Closure under *
- Commutativity of * (Correct answer)
- Associativity of *
- Existence of an identity element
Correct answer: Commutativity of *
A group requires closure, associativity, an identity, and inverses; commutativity is an additional property that defines an abelian (commutative) group, not a group in general.
Question 2: What is the order of the group Z_12 (integers modulo 12 under addition)?
- 6
- 11
- 12 (Correct answer)
- 24
Correct answer: 12
Z_12 has elements {0, 1, 2, ..., 11}, so its order (number of elements) is exactly 12.
Question 3: What is the order of the element 4 in the group Z_12?
- 3 (Correct answer)
- 4
- 6
- 12
Correct answer: 3
The order of element a in Z_n is n/gcd(a, n), so ord(4) = 12/gcd(4, 12) = 12/4 = 3.
Question 4: Which of the following groups is NOT cyclic?
- Z_7
- Z_4
- Z_2 × Z_2 (Correct answer)
- Z_6
Correct answer: Z_2 × Z_2
Z_2 × Z_2 (the Klein four-group) has no element of order 4, so no single element generates the entire group.
Question 5: Lagrange's Theorem states that for a finite group G with subgroup H:
- |H| divides |G| (Correct answer)
- |G| divides |H|
- |H| = |G|/2 always
- |G| = |H|²
Correct answer: |H| divides |G|
Lagrange's Theorem states that the order of any subgroup H of a finite group G must divide the order of G.
Question 6: A subgroup N of a group G is called normal if which condition holds?
- N is cyclic
- gNg⁻¹ = N for all g in G (Correct answer)
- N has prime order
- N equals the center Z(G)
Correct answer: gNg⁻¹ = N for all g in G
A normal subgroup N satisfies gNg⁻¹ = N for every g ∈ G, meaning N is closed under conjugation by any group element.
Question 7: What is the order of the symmetric group S_3?
- 3
- 6 (Correct answer)
- 9
- 12
Correct answer: 6
S_n has order n!, so S_3 has order 3! = 6, corresponding to all permutations of 3 elements.
BMath Bachelor of Mathematics Abstract Algebra and Group Theory 1
Which of the following is NOT a required property for a set G with binary operation * to be a group?