Abstract Algebra

Every group of prime order is abelian.

• True

• False

Let a binary operation ‘*’ be defined on a set A. The operation will be commutative if ________

• 
  a*b=b*a

• (b ο c)*a=(b*a) ο (c*a)

• (a*b)*c=a*(b*c)

• a*b=a

What is an element of the group which can generate the rest of the elements of the group considering the properties that a group must satisfy?

• 
  Identitiy Element

• Inverse Element

• Group Generator

• None of the Above

Which is not a characteristic of a Klein-4 Group?

• 
  It is an Abelian Group.

• 
  The inverse of each element is itself.

• 
  It is a Cyclic Group.

• The product of two elements is the other non-identity element.

A homomorphism f: G→G’ is an isomorphism iff the kernel K is equal to _____.

• {e}       

• G’

• G

• empty set

Consider the binary operations on X, a*b = a+b+4, for a, b ∈ X. It satisfies the properties of _______.

• NOT A GROUP

• ABELIAN GROUP

• GROUP

In a group there must be only __________ identity element.

• 1

• 2

• 3

• 5

Which of the following multiplication tables defined on the set G={a,b,c,d} form a group?

• 
• 
• 
• 

The sets R, Q, Z, 2Z with the usual addition have identity element which is 0. What is the inverse element of these sets for any element (a)?

• 0

• a

• 1

• -a

Which of the following is a group under addition?

• 
  The set of all positive integers.

• The set of all negative integers.

• The set of all odd integers.

• The set of all even integers.