66 3). Show that p is an onto homomorphism, an d describes Rhe L. *b 64. Define

Question

66

3). Show that p is an onto homomorphism, an d describes Rhe L. *b 64. Define K (0,0),(1,1),(22) (a subgroup of 2^ XZs), and prove that z 65. Let G be an abelian group. Define H- [r2:zeG) and KE 66. Without creating homomorphisms, explain why we know that Zq can defined byp (f)-/( As usual Z a group under +3 Prove HG k. be a homp omo- morphic image for a group of order 30 67. Find all of the possible homomorphic images of the group (Zs, +8) ind all of the possible homomorphic images of the group (Zo, +9) Find all of the possible homomorphic images of the group (U(9), 9). Showth is cyclic to help.] 70. Let G be a group with H aG and K a subgroup of G. Prove K/Hnk HK/H

Explanation / Answer

Z4 has a elements of order 4

But 4 does not divide 30.

so preimage of the elements of the order 4 cant divide 30......Or

2nd kind of solution

If there is a such homomorphism then

30 / Order of kernel = 4, we cant find such integral order kernel for any homomorphishm.

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