Show that the order of the group of automorphism of a finite group of order n is a divisor of n!.
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- 4. List all the elements of the subgroupin the group under addition, and state its order.Suppose that G is a finite group. Prove that each element of G appears in the multiplication table for G exactly once in each row and exactly once in each column.Use mathematical induction to prove that if a is an element of a group G, then (a1)n=(an)1 for every positive integer n.
- 15. Prove that on a given collection of groups, the relation of being a homomorphic image has the reflexive property.Exercises 3. Find an isomorphism from the additive group to the multiplicative group of units . Sec. 16. For an integer , let , the group of units in – that is, the set of all in that have multiplicative inverses, Prove that is a group with respect to multiplication.Exercises 30. For an arbitrary positive integer, prove that any two cyclic groups of order are isomorphic.
- Prove that any group with prime order is cyclic.Let G be an abelian group of order 2n, where n is odd. Use Lagranges Theorem to prove that G contains exactly one element of order 2.19. a. Show that is isomorphic to , where the group operation in each of , and is addition. b. Show that is isomorphic to , where all group operations are addition.
- Exercises 18. Suppose and let be defined by . Prove or disprove that is an automorphism of the additive group .27. a. Show that a cyclic group of order has a cyclic group of order as a homomorphic image. b. Show that a cyclic group of order has a cyclic group of order as a homomorphic image.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 .