Prove that a subgroup of a finite abelian group is abelian. Be careful when checking the required conditions.
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Prove that a subgroup of a finite abelian group is abelian. Be careful when checking the required conditions.
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- Exercise 8 states that every subgroup of an abelian group is normal. Give an example of a nonabelian group for which every subgroup is normal. Exercise 8: Show that every subgroup of an abelian group is normal.43. Suppose that is a nonempty subset of a group . Prove that is a subgroup of if and only if for all and .Find a subset of Z that is closed under addition but is not subgroup of the additive group Z.
- 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 2.26. Prove or disprove that if a group has an abelian quotient group , then must be abelian.31. (See Exercise 30.) Prove that if and are primes and is a nonabelian group of order , then the center of is the trivial subgroup . Exercise 30: 30. Let be a group with center . Prove that if is cyclic, then is abelian.