# Groups174What's the most difficult aspect of your life as a mathematician, DianeMaclagan, an assistant professor at Rutgers, was asked. "Trying to provetheorems," she said. And the most fun? "Trying to prove theorems."Exercises1. Prove that the external direct product of any finite number ofsa group. (This exercise is referred to in this chapter.)2. Show that Z, Z, Z, has seven subgroups of order 23. Let G be a group with identity e and let H be a group with iden-tity e Prove that G is isomorphic to Ge} and that H is isomorphic to (eg) H.4. Show that G H is Abelian if and only if G and H are AbelianState the general case.5. Prove or disprove that Z Z is a cyclic group.6. Prove, by comparing orders of elements, that Zg Z, is not iso-morphic to Z, Z.7. Prove that G, G, is isomorphic to G, G. State the generalgroups284case.8. Is Z,Z, isomorphic to Z2? Why?9. Is Z,z, isomorphic to Z,5? Why?10. How many elements of order 9 does Z Z, have? (Do not do thisexercise by brute force.)11. How many elements of order 4 does Z, Z, have? (Do not do thisby examining each element.) Explain why Z Znumber of elements of order 4 as does Zs00000 Z400000ize to the case Z Z.2930has the sameGeneral-313212. Give examples of four groups of order 12, no two of which areisomorphic. Give reasons why no two are isomorphic.13. For each integer n > 1, give examples of two nonisomorphic14. The dihedral group D, of order 2n (n 3) has a subgroup of n ro-morphic to the external direct product of two such groups.of order n2.groups33.tations and a subgroup of order 2. Explain why D, cannot be iso-15. Prove that the group of complex numbers under addition is iso-16. Suppose that G, G2 and H, H2. Prove that G, H G34.35.morphic to RR.36.H,. State the general case.17. If GH is cyclic, prove that G and H are37.case.18. In Z4 Z38.cyclic. State the generafind two subgroups of order 12.30

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Step 1

To decide if the first group (the direct sum) is isomorpbic to Z27,

Step 2

Description of the groups whose isomorphism is under question. The question arises because both (the direct sum as well as Z27 contain the same number of elements:27). We will show that these two groups are NOT isomorphic

Step 3

First observe that the direct su...

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