Let G be an Abelian group and define H to be the set of all elements of G with finite order. Follow the parts below to show that H is a subgroup of G. (a) Suppose x ∈ H. What does this imply about x? (Think about the definition of order and what it means to have finite order.) (b) Suppose y ∈ G. What would you need to show about y in order to show it is in H? (c) Show that H is a nonempty subset of G. (d) Suppose x, y ∈ H where x has order m and y has order n. Show ((xy)^mn) = e. Explain how this implies that the order of xy is at most mn and then explain how this shows that the order of xy is finite. (e) Suppose x ∈ H where x has order m. Show that |x^-1| ≤ m. Explain how this implies that the order of x^-1 is finite. (f) Use your work from the previous parts to show that H is a subgroup of G.
Let G be an Abelian group and define H to be the set of all elements of G with finite order.
Follow the parts below to show that H is a subgroup of G.
(a) Suppose x ∈ H. What does this imply about x? (Think about the definition of order and
what it means to have finite order.)
(b) Suppose y ∈ G. What would you need to show about y in order to show it is in H?
(c) Show that H is a nonempty subset of G.
(d) Suppose x, y ∈ H where x has order m and y has order n. Show ((xy)^mn) = e. Explain how
this implies that the order of xy is at most mn and then explain how this shows that the
order of xy is finite.
(e) Suppose x ∈ H where x has order m. Show that |x^-1| ≤ m. Explain how this implies that
the order of x^-1 is finite.
(f) Use your work from the previous parts to show that H is a subgroup of G.
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