#12 2:00A $3.amazonaws.comChapter 3 Homework1. Prove that in any group, an element and its inverse have the sameorder.2. Let x be in a group G. If x - e and x* -e, prove that x- e and x' e3. Show that Z. -< 3(mod 10) >4. If H and K are subgroups of a group G, show that HÇK is asubgroup of G.5. If #. :aEA are a family of subgroups of the group G, show thatis 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 aretheir 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. LetC(H) - {g€G\gh = hg, VhɛH} Show that C(H)

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Description: #12 2:00A $3.amazonaws.comChapter 3 Homework1. Prove that in any group, an element and its inverse have the sameorder.2. Let x be in a group G. If x - e and x* -e, prove that x- e and x' e3. Show that Z. - 4. If H and K are subgroups of a

12. Consider the group of complex numbers C with addition. Let H=a+bi|a,b∈R, ab≥0. Clearly, the…

Answered: 12. Let "- (a + bi |a.bER, ab a 0}.…

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12. Let "- (a + bi |a.bER, ab a 0}. Determine whether H is a subgroup of the complex numbers C with addition.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.5: Isomorphisms

Problem 10E: Exercises 10. Find an isomorphism from the multiplicative group to the group with multiplication...

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#12

2:00
A $3.amazonaws.com
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.

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