Explanation

The correct answer is: n.

A cyclic permutation of length n is a permutation that shifts each element one position to the right (or left) in a cycle. For example, the cyclic permutation (1 2 3 4 5) maps 1 to 2, 2 to 3, 3 to 4, 4 to 5, and 5 to 1.

The order of a cyclic permutation of length n is n, because it takes n applications of the permutation to return to the original arrangement.

Explanation

The correct answer is: n!.

If X has n elements, then the number of permutations (i.e., arrangements) that can be taken on X is given by the factorial function:

n! = n × (n-1) × (n-2) × ... × 2 × 1

This counts the number of ways to arrange the n elements in a particular order.

Explanation

The group G = {-1, 1, -i, i} is a multiplicative group, and the order of an element a in G is the smallest positive integer n such that a^n = 1.

For -i, we have:

(-i)^2 = -1

(-i)^3 = i

(-i)^4 = 1

So, the order of -i is 4.

Explanation

• The number of left cosets of H in G is equal to the number of right cosets of H in G.
• This is a fundamental result in group theory, known as the "Equivalence of Left and Right Cosets".

Explanation

• Every subgroup of a cyclic group is also cyclic.
• This is because a subgroup of a cyclic group is generated by some power of the generator of the original group.

Explanation

• A cyclic group is a group that can be generated by a single element, i.e., every element in the group can be expressed as a power of that element.
• Cyclic groups are always Abelian, meaning that the group operation is commutative (i.e., the order of elements does not matter).

Explanation

• The correct answer is: a + (-b) ∈ H.
• A nonempty subset H of G is a subgroup of G if and only if it is closed under the group operation + and contains the inverse of each of its elements.

Explanation

The set of nth roots of unity, denoted by {ω₀, ω₁, ..., ωₙ₋₁}, forms a group under the operation of multiplication.

Explanation

The set of integers, denoted by Z, forms an abelian group under the operation of addition (+). This means that the following properties hold: 

• Closure: For any integers a, b, their sum a + b is also an integer.
• Associativity: (a + b) + c = a + (b + c) for any integers a, b, c.
• Commutativity: a + b = b + a for any integers a, b.
• Identity: There exists an identity element, which is 0, such that a + 0 = a for any integer a.
• Inverse: For each integer a, there exists an inverse element, which is -a, such that a + (-a) = 0.

The set of integers does not form a group under multiplication, because the inverse property does not hold (except for 1 and -1). Division is not even a binary operation on the set of integers, because it is not defined for all pairs of integers (e.g., 1/2 is not an integer).

Therefore, the correct answer is Addition.

Explanation

In a group (G, *), the binary operation * is defined in such a way that the result of combining any two elements is always an element in the group. Therefore, if a, b ∈ G, then a * b ∈ G.