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Midterm

# MATH546 South Carolina 546 f04 2Exam

Department
Mathematics
Course Code
MATH 546
Professor
All
Study Guide
Midterm

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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.
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 aGwith a6=id,but a
2=id.
5. (7 points) Give an example of a nite 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 g1hg H
for all gGand hH. Fix an element gG.ProvethatgH =Hg ,
where gH is the LEFT coset
gH ={gh |hH}
and Hg is the RIGHT coset
Hg ={hg |hH}.