14.1. Unique Identity Element

Try these first

1. 1.

   What is the identity element in a group?

   Hint

   It is represented by the letter e in many mathematical contexts.

2. 2.

   Define an inverse element.

   Hint

   Think about reversing an operation.

3. 3.

   What does the identity element do in a group?

   • True
   • False

   Hint

   Consider the definition of identity.

4. 4.

   Can a group have multiple identity elements?

   • Yes
   • No
   • Only under special conditions

   Hint

   Think about the properties of identity.

5. 5.

   Show that in any finite group, a generator's order must equal the group's order.

   Hint

   Think about all elements and how repetition constrains output.

6. 6.

   In a cyclic group of odd order, show every element must also have an odd order.

   Hint

   Consider how inverses match within powers.