Test: Group Theory - Computer Science Engineering (CSE) MCQ

An algebraic structure (P,*) is called a semigroup if a*(b*c) = (a*b)*c for all a,b,c belongs to S or the elements follow associative property under “*”. (Matrix,*) and (Set of integers,+) are examples of semigroup.

A monoid(B,*) is called Group if to each element there exists an element c such that (a*c)=(c*a)=e. Here e is called an identity element and c is defined as the inverse of the corresponding element.

The set of two M*M non-singular matrices form a group under matrix multiplication operation. Since matrix multiplication is itself associative, it holds associative property.

The correct answer is d) 4.

A mathematical group must satisfy the following four properties:

1. Closure: For any two elements aaa and bbb in the group, their operation a∗ba \ast ba∗b must also be in the group.
2. Associativity: The operation must be associative, i.e., (a∗b)∗c=a∗(b∗c)(a \ast b) \ast c = a \ast (b \ast c)(a∗b)∗c=a∗(b∗c) for all a,b,ca, b, ca,b,c in the group.
3. Identity: There must exist an identity element eee in the group such that a∗e=e∗a=aa \ast e = e \ast a = aa∗e=e∗a=a for all aaa in the group.
4. Inverse: For every element aaa in the group, there must exist an inverse element a−1a^{-1}a−1 such that a∗a−1=a−1∗a=ea \ast a^{-1} = a^{-1} \ast a = ea∗a−1=a−1∗a=e, where eee is the identity element.

These four properties are essential for a set and operation to qualify as a group.

The set of complex numbers {1, i, -i, -1} under multiplication operation is a cyclic group. Two generators i and -i will covers all the elements of this group. Hence, it is a cyclic group.

A non empty set A is called an algebraic structure w.r.t binary operation “*” if (a*b) belongs to S for all (a*b) belongs to S. Therefore “*” is closure operation on ‘A’.

A Semigroup (S,*) is defined as a monoid if there exists an element e in S such that (a*e) = (e*a) = a for all a in S. This element is called identity element of S w.r.t *.

A group (M,*) is said to be abelian if (x*y) = (x*y) for all x, y belongs to M. Thus Commutative property should hold in a group.

A singular element can generate a cyclic group. Every element of a cyclic group is a power of some specific element which is known as a generator ‘g’.

A cyclic group is always an abelian group but every abelian group is not a cyclic group. For instance, the rational numbers under addition is an abelian group but is not a cyclic one.