13.1. Definition of Groups

Try these first

1. 1.

   What is the definition of a group?

   Hint

   Remember the four key properties.

2. 2.

   Can a set of non-negative integers form a group under addition?

   Hint

   Consider what happens when you try to find the inverse of an element.

3. 3.

   What defines a group in mathematics?

   • A set with addition
   • A set with a binary operation satisfying certain axioms
   • Any collection of numbers

   Hint

   Think about the requirements needed for a group.

4. 4.

   Is the identity element unique in a group?

   • True
   • False

   Hint

   Consider the definition of identity in groups.

5. 5.

   Consider the set of rational numbers under addition. Does it form a group? Justify your answer using the group axioms.

   Hint

   Check each axiom systematically.

6. 6.

   Propose a new operation on a set of your choice and prove whether or not it forms a group.

   Hint

   Define your operation clearly and work through examples.