Department

MathematicsCourse Code

MATH 546Professor

AllStudy Guide

MidtermThis

**preview**shows half of the first page. to view the full**1 pages of the document.**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}.

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