13.6.1. Abelian Groups

Try these first

1. 1.

   Define an Abelian group.

   Hint

   Recall the definition of commutativity.

2. 2.

   Give an example of an Abelian group.

   Hint

   Think of the number line.

3. 3.

   What is an Abelian group?

   • A group where operations are non-commutative
   • A group where operations are commutative
   • A random set of numbers

   Hint

   Remember the distinction between commutative and non-commutative.

4. 4.

   Is the group of integers under multiplication an Abelian group?

   • True
   • False

   Hint

   Consider numbers like 0.

5. 5.

   Show that for any Abelian group (G, *), if a, b, and c belong to G, then a * (b * c) = (a * b) * c.

   Hint

   Start by applying the definition of associativity.

6. 6.

   Given two Abelian groups, G1 and G2, prove that their direct product G1 x G2 is also an Abelian group.

   Hint

   Check how operations interact in the composite set.