Test: Group Theory - 3 - Mathematics MCQ

The number of generators in a cyclic group of order 10 can be determined by identifying the elements that can generate the entire group through their powers.

• A cyclic group of order 10 has elements represented as integers modulo 10: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
• To be a generator, an element must be relatively prime to 10. This means the greatest common divisor (GCD) of the element and 10 must be 1.
• The integers that meet this criterion are:
  • 1 (GCD(1, 10) = 1)
  • 3 (GCD(3, 10) = 1)
  • 7 (GCD(7, 10) = 1)
  • 9 (GCD(9, 10) = 1)
• Thus, the generators of the cyclic group of order 10 are 1, 3, 7, and 9.

In conclusion, there are four generators for a cyclic group of order 10.

A group of prime order must be cyclic and every cyclic group is abelian. Then we can show that φ: G → G s.t. φ(x) = xⁿ is an isomorphism if 0(G) and n and are co-prime.

• Intersection H₁ ∩ H₂: Always a subgroup. It contains elements common to both H₁ and H₂, satisfying closure and inverses.
• Union H₁ ∪ H₂: Not necessarily a subgroup unless one is contained in the other.
• Product H₁H₂: Not always a subgroup unless specific conditions are met.
• None of these: Incorrect since the intersection is always a subgroup.

Here, 6 = 2 x 3

The correct answer is:

1. infinity

In the additive group of integers, the order of an element aaa (where a≠0 ) refers to the smallest positive integer nnn such that n⋅a=0 However, for any non-zero integer aaa, there is no positive integer nnn that satisfies this equation because adding aaa to itself any number of times will never result in 0. Therefore, the order of any non-zero element in this group is infinite.

The set of all nth roots of unity from a finite abelian group of order n can be understood in the context of different operations.

• Addition: In this operation, the roots combine to form new elements, but they do not maintain the properties of the roots of unity.
• Subtraction: Similar to addition, subtracting roots does not preserve the unity characteristic.
• Multiplication: This is the key operation. When you multiply the nth roots of unity, you obtain another nth root of unity, preserving the structure of the group.
• Division: While division can be performed, it does not yield a consistent result that maintains the property of being a root of unity.

The multiplication of these roots is significant because it allows us to remain within the set of nth roots of unity, whereas addition and subtraction do not. Therefore, the most appropriate operation to consider with respect to the nth roots of unity is multiplication.

we have o(G) = 6 and prime to 6 are 1 and 5