Abstract Algebra | MCQs

Explanation

The additive inverse of a number is what you add to a number to get zero.

In this context, 1 and -1 are additive inverses of each other, satisfying:

3 + (-1) = 2 and 1 - 3 = -2, showing the concept revolves around opposites in addition.

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

• A group of prime order (a group with a prime number of elements) is always a cyclic group.
• This is a fundamental result in group theory, known as Lagrange's theorem.

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.