Problem 5.5.1. Let G and H by two groups, and let g = G and hЄ H be elements of finiteorder. Find the order of (g, h) in the group G × H, expressed in terms of |g|and h.5.2. Let m,n Є Z, and let us write [a]m and [a] for the equivalence classes ofa = Z in Z/mZ and Z/nZ, respectively. Show that ([1]m, [1] n) generates thegroup Z/mZxZ/nZ if and only if m and n are relatively prime.

Answered: Image /qna-images/answer/92152b5f-5240-4d2d-b103-9139503d8e1e.jpg

Answered: Problem 5. 5.1. Let G and H by two…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.4: Cosets Of A Subgroup

Problem 12E: Let H and K be subgroups of a group G and K a subgroup of H. If the order of G is 24 and the order...

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Let H and K be subgroups of a group G and K a subgroup of H. If the order of G is 24 and the order of K is 3, what are all the possible orders of H?
Find two groups of order 6 that are not isomorphic.
If a is an element of order m in a group G and ak=e, prove that m divides k.
12. Find all homomorphic images of each group in Exercise of Section. 18. Let be the group of units as described in Exercise. For each value of, write out the elements of and construct a multiplication table for . a. b. c. d.
Exercises 3. Find an isomorphism from the additive group to the multiplicative group of units . Sec. 16. For an integer , let , the group of units in – that is, the set of all in that have multiplicative inverses, Prove that is a group with respect to multiplication.
42. For an arbitrary set , the power set was defined in Section by , and addition in was defined by Prove that is a group with respect to this operation of addition. If has distinct elements, state the order of .
15. Prove that on a given collection of groups, the relation of being a homomorphic image has the reflexive property.
Write 20 as the direct sum of two of its nontrivial subgroups.
Let H and K be arbitrary groups and let HK denotes the Cartesian product of H and K: HK=(h,k)hHandkK Equality in HK is defined by (h,k)=(h,k) if and only if h=h and k=k. Multiplication in HK is defined by (h1,k1)(h2,k2)=(h1h2,k1k2). Prove that HK is a group. This group is called the external direct product of H and K. Suppose that e1 and e2 are the identity elements of H and K, respectively. Show that H=(h,e2)hH is a normal subgroup of HK that is isomorphic to H and, similarly, that K=(e1,k)kK is a normal subgroup isomorphic to K. Prove that HK/H is isomorphic to K and that HK/K is isomorphic to H.
Exercises 3. Find the order of each element of the group in Example of section. Example 3. We shall take and obtain an explicit example of . In order to define an element of , we need to specify , , and . There are three possible choices for . Since is to be bijective, there are two choices for after has been designated, and then only one choice for . Hence there are different mappings in .
Label each of the following statements as either true or false. Two groups can be isomorphic even though their group operations are different.
10. Prove that in Theorem , the solutions to the equations and are actually unique. Theorem 3.5: Equivalent Conditions for a Group Let be a nonempty set that is closed under an associative binary operation called multiplication. Then is a group if and only if the equations and have solutions and in for all choices of and in .

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