Test: Group Theory - 12 - Mathematics MCQ

A relation (34×78)×57=57×(78×34) is given, and we are asked to determine its property.
This relation exhibits the associative property since changing the grouping of the numbers in the operation does not change the result.

Let (A7,⊗7)=({1,2,3,4,5,6},⊗7) be a group. It has two subgroups X={1,3,6} and Y={2,3,5}. We are asked to find the order of the union of subgroups X and Y.
The order of a group is the number of elements in the group. The union of two sets is the set of all distinct elements from both sets. So, the order of the union of subgroups X and Y is the total number of distinct elements in X∪Y.
X∪Y={1,2,3,5,6}
The order of X∪Y is 5.

Explanation: As it is an intersection so the order must divide both K and L. Here 3, 6, 30 does not divide 14. But 5 must be the order of the group as it divides the order of intersection of K and L as well as the order of the group.

Two statements B1​:({0,1,2,…,(n−1)},xm​) and B₂:({0,1,2,…,n},x_n)B2​:({0,1,2,…,n},xn​) are given. Both B₁​ and B₂​ are considered to be semigroups.
These statements describe sets equipped with a multiplication operation modulo m and n, respectively. Since there is no mention of an identity element or inverses, and the operation is only associative, both B1​ and B2​ are considered semigroups.

Explanation: For identity e, a+e=e+a=e, a*e = a+e+ae = a => e=0 and e+a = e+a+ea = a => e=0. So e=0 will be identity, for e to be identity, a*e = a ⇒ a+e+ae = a ⇒ e+ae = 0 and e(1+a) = 0 which gives e=0 or a=-1. So, when a = -1, no identity element exist as e can be any value in that case.

Consider the binary operation on X, a∗b=a+b+4, for a,b∈X. We are asked to determine the properties satisfied by this operation.
This operation satisfies the properties of an abelian group since it is closed under addition, associative, has an identity element (0), and every element has an inverse.

A group G, {0},+) under addition operation satisfies which of the following properties?
The group G satisfies the properties of closure, associativity, inverse, and identity since there is only one element (0) in the set, making closure and identity trivial, and associativity and inverses are always satisfied for any group.

Let G be a finite group with two subgroups M and N such that M∣=56 and ∣N∣=123. We are asked to determine the value of ∣M∩N∣.
The order of the intersection of subgroups M and N is ∣M∩N∣=1 because the identity element is common to both subgroups.

The set of even natural numbers, {6,8,10,12,…}, is closed under addition operation. We are asked to identify the property it satisfies.
This set satisfies the closure property under addition since the sum of any two even numbers is also even.

If (M,∗) is a cyclic group of order 73, then the number of generators of G is equal to 72.
For a cyclic group of order n, there are ϕ(n) generators, where ϕ is Euler's totient function. For n=73, ϕ(73)=72.

The multiplicative identity of a set of numbers is the number that, when multiplied by any other number in the set, leaves that number unchanged. For natural numbers, this identity is 1. When any natural number is multiplied by 1, the result is the original number.

In group theory, a group is a set equipped with a binary operation that satisfies four fundamental properties - closure, associativity, identity element, and inverse element. A group must have exactly one identity element. This is because for any element 'a' in the group, there must exist an inverse 'b' such that the product 'a * b' (or 'b * a') equals the identity element. If there were more than one identity element, this property would be violated, and the set would not form a group.

A singular matrix is a square matrix that does not have a unique solution, meaning its determinant is zero. Matrices with determinants equal to zero do not have multiplicative inverses. In contrast, non-singular matrices have unique solutions and possess multiplicative inverses.

The identity element of a group is an element such that combining it with any other element in the group leaves the other element unchanged. The inverse element is the element that, when combined with another element, produces the identity element. Therefore, the identity element of a group has an inverse element.

Using the properties of matrix inverses, the expression simplifies to (AB)C, which is equivalent to ABC. Since matrix multiplication is associative, the final result is BC.

An idempotent matrix is a matrix that, when multiplied by itself, gives the same matrix. If X is idempotent and nonsingular, it implies that X * X = X and X has an inverse. The only matrix that satisfies these conditions is the identity matrix.

Lie groups are mathematical structures that describe continuous symmetries. In the context of the Standard Model in physics, Lie groups play a crucial role in representing the symmetries of elementary particle interactions.

If the sum of elements in each row of a matrix is zero, it implies that the matrix's rows are linearly dependent. Such matrices are singular because they do not have full rank, and their determinant is zero.

An idempotent element in a monoid is one that, when combined with itself, produces the same element. Mathematically, this is expressed as a² = a * a = a.

A monoid is a semigroup that has an identity element. In a monoid, there exists an element (the identity) such that combining it with any other element in the set leaves the other element unchanged under the specified binary operation.