2(a) 9. Let G be a group. Show that Z(G) =

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Chapter2: Second-order Linear Odes
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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.
Transcribed Image Text: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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