Math 420/620

Problem Set 2

 

 

  1. Let be a finite group (say
).
  1. Show any element of has finite order.
  2. Show that
.

 

(What does this tell you about the Rubik’s cube question that I asked on the first day of class?)

 

 
  1. Show that any group of exponent 2 is abelian.

    2.  

  2. In this problem we will show that any finite group generated by two (distinct) involutions (elements of order 2) is a dihedral group.

 

  1. Assume that is finite and is generated by and , two distinct elements of order 2. As is assumed finite, what can be said about the order of the element ?
  2. Let and from the definition of in class. What can you say about our group generated about and and the dihedral group
?

 

  1. Show if and are groups then the group is abelian if and only if both and are abelian.

    2.  

  2. We define the center of a group to be (The center of a group is the collection of elements that commute with any element of Note that the identity is always in the center). Compute
.