13.2. Group Axioms

Try these first

1. 1.

   What is the first axiom of a group?

   Hint

   Consider what happens when you combine two numbers from the set.

2. 2.

   Is zero the identity element in the group of integers under addition?

   Hint

   Think about adding zero to any number.

3. 3.

   What does the closure property entail?

   • The operation can produce an element outside the set
   • The operation must produce an element within the set
   • It is an optional property

   Hint

   Think about the results of combining elements from a set.

4. 4.

   True or False: An identity element must be unique within a group.

   • True
   • False

   Hint

   Consider the implications of having multiple identity elements.

5. 5.

   Consider the set of all integers. Prove whether it forms a group under subtraction.

   Hint

   Think about operations and how subtraction behaves compared to addition.

6. 6.

   Given a set of matrices with a defined matrix addition, explain whether it can form a group and why.

   Hint

   Do matrices behave like numbers in terms of addition?