Prove that any two Sylow p-subgroups of a group G are conjugate.

Q: Prove that the subset of elements of finite order in an Abelian groupforms a subgroup. (This…

A: To Prove: The subset of elements of finite order in an abelian group forms a subgroup.

Q: Find a proper subgroup of the group of integers Z under addition and prove that this subgroup is…

A: Consider 2 ℤ = {an / n∈ ℤ }This is a subgroup of ℤ .claim : 2ℤ is isomorphic to ℤ.…

Q: Prove that every subgroup of an abelian group is abelian

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Q: Show that the group of automorphism of a cyclic group of degree four is of order two.

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Q: Prove that every group of order 675 has a normal Sylow 5-subgroup.

A: Theorem: If number of Sylow p-subgroup of a group G is unique, then that subgroup is normal. Let G…

Q: Prove that any subgroup H (of a group G) that has index 2 (i.e. only 2 cosets) must be normal in G

A: To show that H is a normal subgroup we have to show that every left coset is also a right coset. We…