In Exercises 1- 9, let
Let
Figure 4.2
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Elements Of Modern Algebra
- In Exercises 1- 9, let be the given group. Write out the elements of a group of permutations that is isomorphic to, and exhibit an isomorphism from to this group. 9. Let be the octic group .arrow_forward12. 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.arrow_forwardShow that a group of order 4 either is cyclic or is isomorphic to the Klein four group e,a,b,ab=ba.arrow_forward
- In Exercises 1- 9, let G be the given group. Write out the elements of a group of permutations that is isomorphic to G, and exhibit an isomorphism from G to this group. Let G be the quaternion group { 1,i,j,k }arrow_forwardExercises List all the elements of the alternating group A3, written in cyclic notation.arrow_forwardExercises 10. Find an isomorphism from the multiplicative group to the group with multiplication table in Figure . This group is known as the Klein four group. Figure Sec. 16. a. Prove that each of the following sets is a subgroup of , the general linear group of order over . Sec. 3. Let be the Klein four group with its multiplication table given in Figure . Figure Sec. 17. Show that a group of order either is cyclic or is isomorphic to the Klein four group . Sec. 16. Repeat Exercise with the quaternion group , the Klein four group , and defined byarrow_forward
- 11. Find all normal subgroups of the alternating group .arrow_forwardLet a,b,c, and d be elements of a group G. Find an expression for (abcd)1 in terms of a1,b1,c1, and d1.arrow_forwardLet 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?arrow_forward
- Find a subset of Z that is closed under addition but is not subgroup of the additive group Z.arrow_forwardFind two groups of order 6 that are not isomorphic.arrow_forwardIn Exercises , is a normal subgroup of the group . Find the order of the quotient group . Write out the distinct elements of and construct a multiplication table for . 2. The octic group ; .arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,