Let G = Z9 X Z12 X Z16- (a) Find all subgroups of G of order 144.
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please show the steps on to get all of the subgroups
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- Exercises 31. Let be a group with its center: . Prove that if is the only element of order in , then .27. Suppose is a normal subgroup of order of a group . Prove that is contained in , the center of .18. If is a subgroup of the group such that for all left cosets and of in, prove that is normal in.
- Let G be an abelian group. For a fixed positive integer n, let Gn={ aGa=xnforsomexG }. Prove that Gn is a subgroup of G.Suppose G1 and G2 are groups with normal subgroups H1 and H2, respectively, and with G1/H1 isomorphic to G2/H2. Determine the possible orders of H1 and H2 under the following conditions. a. G1=24 and G2=18 b. G1=32 and G2=40Let be a subgroup of a group with . Prove that if and only if .
- 14. Let be an abelian group of order where and are relatively prime. If and , prove that .22. If and are both normal subgroups of , prove that is a normal subgroup of .Let be a group of order , where and are distinct prime integers. If has only one subgroup of order and only one subgroup of order , prove that is cyclic.