11. Let H be a subgroup of R under addition. Let K- (2" |a€H} , Show that K is a subgroup of R* under multiplication.
Unitary Method
The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products.
Speed, Time, and Distance
Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.
Profit and Loss
The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.
Units and Measurements
Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.
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Chapter 3 Homework
1. Prove that in any group, an element and its inverse have the same
order.
2. Let x be in a group G. If x - e and x* -e, prove that x- e and x' e
3. Show that Z. -< 3(mod 10) >
4. If H and K are subgroups of a group G, show that HÇK is a
subgroup of G.
5. If #. :aEA are a family of subgroups of the group G, show that
is a subgroup of G.
6. If G is a group and a is an element of G. show that C(a) - C(a")
7. Use the following Cayley table for the Group G to answer 7A, 7B,
and 7C.
7A. Given the group G above, find C(a) for all a in G:
7B. Find the Z(G) for the group above.
7C. Find the order of each element in the group G above.
8. Give an example of a group G where the set of all elements that are
their own inverses does NOT form a subgroup.
(a)
9. Let G be a group. Show that Z(G) =
10. Let G be a group and H a subgroup of G. Let
C(H) - {g€G\gh = hg, VhɛH} Show that C(H)<G.
11. Let H be a subgroup of R under addition. Let K- {2" | a€H} , Show
that K is a subgroup of R* under multiplication.
12. Let "- {a + bi |a,bƐR, ab z 0} . Determine whether H is a subgroup of the
complex numbers C with addition.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc074fc7e-4e48-46f1-bfc4-aee6ba4e6262%2F324e2fad-252d-4a7d-88c0-24beaa0ae7a2%2Fc0uuve_processed.jpeg&w=3840&q=75)
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