Group Theory | MCQs

Explanation

A group generated by one of its elements is called a cyclic group.

This means that the group can be generated by repeatedly applying the group operation to a single element, called the generator.

In other words, all elements of the group can be expressed as powers of the generator.

Explanation

1. Commutative Law: In an Abelian group, for all elements $a$ and $b$ in the group, the operation satisfies $a\cdot b=b\cdot a$.

2. Named after Niels Henrik Abel: The term "Abelian" is derived from the name of the mathematician Niels Henrik Abel.

Explanation

Explanation

Explanation

Given the operation a * b = a + b + 1:

Let's find the identity element 'e' such that a * e = a.

a * e = a + e + 1 = a

This implies e + 1 = 0, so e = -1.

Now, to find the inverse 'x' of 'a' such that a * x = e = -1:

a * x = a + x + 1 = -1

This implies a + x = -2.

x = -2 - a

x = -a - 2

Explanation

Every subgroup of a cyclic group is also cyclic.

This is a key result in group theory under abstract algebra.

Explanation

To find the inverse of 3 in modulo 7, we need to find a number x such that:

3x ≡ 1 (mod 7)

We can try multiplying 3 by each number from 1 to 6:

3 × 1 = 3

3 × 2 = 6

3 × 3 = 9 ≡ 2 (mod 7)

3 × 4 = 12 ≡ 5 (mod 7)

3 × 5 = 15 ≡ 1 (mod 7)

So, the inverse of 3 in modulo 7 is 5.

Explanation

The identity element e should satisfy ae = ea = a for all a in G.

Here, e = 6 works since 26=2, 46=4, 66=6, 86=8.

Explanation

The set of integers, Z, forms a group under addition.

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".