Groups, Subgroups and Cyclic Groups - One Line Questions

1. Consider the group of integers modulo 10 under addition (Z10, +). Which of the following is NOT a subgroup? — {0, 3, 6, 9, 1, 4, 7}
2. Let G = {1, -1, i, -i} be the group of fourth roots of unity under multiplication. Which of the following is a subgroup? — {1, -1}
3. Let G be a group and H a subgroup. If 'a' is an element of G, what is the left coset 'aH'? — {ah | h in H}
4. Which element generates the cyclic group Z4 under addition? — 1
5. What is the identity element in the group of integers under addition? — 0
6. What is the order of the element 3 in the cyclic group Z6 under addition? — 2
7. What is the order of the cyclic group Z5 under addition? — 5
8. If o(a) = n, what is the order of the cyclic subgroup generated by 'a'? — n
9. What is the order of the group of integers modulo n (Zn, +)? — n
10. What is the order of the element 1 in the group of non-zero real numbers under multiplication? — 1
11. Consider the group of integers modulo 7 under addition (Z7, +). What is the order of the element 4? — 7
12. What is the order of the element 5 in the group of integers modulo 10 under addition? — 2
13. What is the order of the element 2 in the group Z12 under addition? — 6
14. What is the order of the subgroup {0, 3, 6, 9} in the group of integers modulo 12 under addition? — 4
15. What is the inverse of element 'a' in the group of non-zero rational numbers under multiplication? — 1/a
16. What is a cyclic group? — A group that can be generated by a single element.
17. What is the relationship between subgroups of a cyclic group? — All subgroups are cyclic.
18. What is a subgroup? — A subset of a group that is itself a group under the same operation.
19. Which property of a group states that for any elements a, b, and c in the group, (a * b) * c = a * (b * c)? — Associativity
20. Let G be a group. For which property is the subset {e} (where e is the identity) always a subgroup? — Closure
21. Which property is NOT required for a set with a binary operation to be a group? — Commutativity
22. What is the defining property of a group in abstract algebra? — Closure, Associativity, Identity, Inverse
23. Which of the following is a property of cyclic groups? — Every cyclic group is abelian.
24. Let G be a cyclic group of order n. How many subgroups does G have? — Exactly one subgroup for each divisor of n.
25. Which operation must be closed under a group (G, *)? — For all a, b in G, a * b is in G
26. Which of the following is a necessary and sufficient condition for a non-empty subset H of a group G to be a subgroup? — For all a, b in H, a * b^(-1) is in H.
27. If G is a cyclic group and g is a generator, which element is the identity element? — g^0
28. What are the trivial subgroups of any group G? — G and the set containing only the identity element.
29. If a group G is cyclic and generated by element 'g', what is the inverse of g^k? — g^(-k)
30. If a subgroup H has index [G:H] = 2 in a group G, what can be said about H? — H is always a normal subgroup.
31. If G is a cyclic group generated by 'g', and H is a subgroup of G, what can be said about H? — H is always cyclic.
32. Which theorem states that the order of a subgroup must divide the order of the group? — Lagrange's Theorem
33. In a cyclic group G generated by 'g', what is the order of the element g^k where n = order(g)? — n / gcd(n, k)
34. What is the set of all elements 'g' in a group G such that g*x = x*g for all x in G called? — The center of G
35. What is the smallest group that contains all cyclic groups of order n? — The cyclic group of order n.
36. Which of the following is a cyclic group under addition? — The group of integers modulo n (Zn, +)
37. What is the index of a subgroup H in a group G, denoted by [G:H]? — The number of distinct left (or right) cosets of H in G.
38. What is the order of a subgroup? — The number of elements in the subgroup.
39. If G is a finite group and 'a' is an element of G, which of the following is always true? — The order of 'a' divides the order of G.
40. Which of the following is always a subgroup of any group G? — The set containing only the identity element.
41. Which of the following sets forms a group under multiplication? — The set of non-zero rational numbers
42. Which of the following is an example of a cyclic group under multiplication? — The set of nth roots of unity
43. Consider the group of integers under addition (Z, +). Which of the following is a subgroup? — The set of even integers (2Z)
44. Which of the following sets forms a group under addition? — The set of all integers (Z)
45. What is the order of the element 'a' in a group G, denoted by o(a)? — The smallest positive integer n such that a^n = e (identity).
46. What is the identity element for the operation in the group of invertible n x n matrices under matrix multiplication? — The identity matrix (I_n)
47. Is the group of non-zero rational numbers under multiplication a cyclic group? — No
48. Let H be a subgroup of G. If H is cyclic, is G necessarily cyclic? — No
49. Consider the group of symmetries of an equilateral triangle (D3). Is it a cyclic group? — No, it is not cyclic because it is not abelian.
50. Is the set of positive integers under addition a group? Why or why not? — No, it lacks inverse elements.
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