15.2.2. Characterization for Subgroups

Try these first

1. 1.

   Define a subgroup.

   Hint

   Think about the interactions of a group with its subset.

2. 2.

   What is the closure property?

   Hint

   What happens to elements when combined?

3. 3.

   Which of the following must be true for a subset to be a subgroup?

   • It must be non-empty
   • It must contain the inverse of each element
   • It must fulfill closure property
   • All of the above

   Hint

   Think about group properties.

4. 4.

   True or False: If a group is finite, every non-empty subset is a subgroup.

   • True
   • False

   Hint

   Consider closure and inverse properties.

5. 5.

   Given a group of symmetries of a square, determine a valid subgroup along with a proof using closure and inverses.

   Hint

   Visualize the square and rotate it to see the outcomes.

6. 6.

   Design a finite group of order 30 and demonstrate Lagrange's theorem by detailing its subgroups.

   Hint

   Use cyclical permutations to visualize group elements.