Math 546, Exam 2, Fall, 2004
The exam is worth 50 points.
Write your answers as legibly as you can on the blank sheets of paper provided.
Use only one side of each sheet. Take enough space for each problem. Turn in
your solutions in the order: problem 1, problem 2, ... ; although, by using enough
paper, you can do the problems in any order that suits you.
If I know your e-mail address, I will e-mail your grade to you. If I don’t already
know your e-mail address and you want me to know it, then send me an e-mail.
I will leave your exam outside my office TOMORROW by about 5PM, you may
pick it up any time between then and the next class.
I will post the solutions on my website at about 4:00 PM today.
1. (6 points) Define “subgroup”. Use complete sentences.
2. (6 points) Define the “center of a group”. Use complete sentences.
3. (6 points) STATE Lagrange’s Theorem.
4. (7 points) Let Gbe a finite group with an even number of elements. Prove
that there must exist an element a∈Gwith a6=id,but a
2=id.
5. (7 points) Give an example of a finite group Gand a proper subgroup Hof
G, with Hnot a cyclic group.
6. (6 points) Let Gbe a group of order pq where pand qare prime numbers.
Prove that every proper subgroup of Gis cyclic.
7. (6 points) Let gbe an element of the group G. Suppose that Ghas order n.
Prove that gn=id.
8. (6 points) (6 points) Let Hbe a subgroup of a group. Suppose that g−1hg ∈H
for all g∈Gand h∈H. Fix an element g∈G.ProvethatgH =Hg ,
where gH is the LEFT coset
gH ={gh |h∈H}
and Hg is the RIGHT coset
Hg ={hg |h∈H}.