MATH 546 Midterm: MATH546 South Carolina 546 94 1 nospace
Math 546, Exam 1, Fall , 1994
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There are 7 problems on 4 pages. The exam is worth a total of 50 points. Problem
1 is worth 8 points. The other problems are worth 7 points each.
1. TRUE or FALSE. (If true, PROVE it. If false, give a COUNTER EXAMPLE.)
If Hand Kare subgroups of a group G, then the intersection H∩Kis also
a subgroup of G.
2. TRUE or FALSE. (If true, PROVE it. If false, give a COUNTER EXAMPLE.)
If Hand Kare subgroups of a group G, then the union H∪Kis also a
subgroup of G.
3. Let Gbe an abelian group with identity element e.Let
H={x∈G|x
2=e}.
Prove that His a subgroup of G.
4. Let Gbe a group with identity element e. Suppose that a,b,and care
elements of Gwith c∗b∗a=e.Provethatb∗a∗cis also equal to e.
5. Let R∗represent the set of nonzero real numbers. Define a binary operation ∗
on R∗by a∗b=b/a .Is(R
∗
,∗) a group? If so prove it. If not, show why not.
6. Let Gbe a group. Let
H={x∈G|xy =yx for all y∈G}.
Prove that His a subgroup of G.
7. Let Gbe a group with identity element e. Suppose that x2=efor all x∈G.
Prove that Gis an abelian group.
Document Summary
The exam is worth a total of 50 points. The other problems are worth 7 points each: true or false. (if true, prove it. If h and k are subgroups of a group g , then the intersection h k is also a subgroup of g : true or false. (if true, prove it. If h and k are subgroups of a group g , then the union h k is also a subgroup of g : let g be an abelian group with identity element e . H = {x g | x2 = e}. Suppose that a , b , and c are elements of g with c b a = e . Prove that b a c is also equal to e : let r represent the set of nonzero real numbers. De ne a binary operation on r by a b = b/a . If not, show why not: let g be a group.
