Let G be a group of order 150 = 2 × 3 × 52. Assume N G with |N| = 6. Prove that N is the only. subgroup of order 6 in G. Hint: Draw a diagram, and use a proof by contradiction.
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Let G be a group of order 150 = 2 × 3 × 52. Assume N G with |N| = 6. Prove that N is the only. subgroup of order 6 in G. Hint: Draw a diagram, and use a proof by contradiction.
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- 15. Assume that can be written as the direct sum , where is a cyclic group of order . Prove that has elements of order but no elements of order greater than Find the number of distinct elements of that have order .Let G be a group of order pq, where p and q are primes. Prove that any nontrivial subgroup of G is cyclic.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.