abstract algebra: a study guide for beginners - Northern Illinois ...

30 CHAPTER 3. GROUPS
MORE PROBLEMS: §3.2
41.† Let G be a group, and let a ∈ G, with a �= e. Prove or disprove these statements.
(a) The element a has order 2 if and only if a 2 = e.
(b) The element a has order 3 if and only if a 3 = e.
(c) The element a has order 4 if and only if a 4 = e.
42. In each of the groups O, P , and Q in Tables 3.1, 3.2, and 3.3 in the Study Guide, find
〈b〉 and 〈ab〉. That is, find the cyclic subgroups generated by b and ab.
43.† Is {(x, y) ∈ R 2 | y = x 2 } a subgroup of R 2 ?
44.† Is {(x, y) ∈ R 2 | x, y ∈ Z} a subgroup of R 2 ?
45. Let G = C[0, 1], the group defined in Problem 3.1.45. Let Pn denote the subset of
functions in G of the form anx n + . . . + a1x + a0, where the coefficients ai all belong
to R. Prove that Pn is a subgroup of G.
46. Let G = C[0, 1], the group defined in Problem 3.1.45. Let X be any subset of [0, 1].
Show that {f ∈ C[0, 1] | f(x) = 0 for all x ∈ X} is a subgroup of G.
47. Show that the group T defined in Problem 3.1.47 is a subgroup of Sym(R 2 ).
48. In G = S5, let H = {σ ∈ S5 | σ(5) = 5}. Find |H|; show that H is a subgroup of G.
49. Let G be a group, and let H, K be subgroups of G. Show that H ∩ K is a subgroup
of K.
50. Let G be a group, and let H, K be subgroups of G.
(a) Show that H ∪ K is a subgroup of G if and only if either H ⊆ K or K ⊆ H.
(b) Show that a group cannot be the union of 2 proper subgroups. Give an example
to show that it can be a union of 3 proper subgroups.
51. Let G be a group, let H be any subgroup of G, and let a be a fixed element of G.
Define aHa −1 = {g ∈ G | g = aha −1 for some h ∈ H}. Show that aHa −1 is a
subgroup of G.
52.† Let G be a group, with a subgroup H ⊆ G. Define N(H) = {g ∈ G | gHg −1 = H}.
Show that N(H) is a subgroup of G that contains H.
53. Let G be a group.
(a) Show that if H1 and H2 are two different subgroups of G with |H1| = |H2| = 3,
then H1 ∩ H2 = {e}.
(b) Show that G has an even number of elements of order 3.
(c) Show that the number of elements of order 5 in G must be a multiple of 4.
(d) Is the number of elements of order 4 in G a multiple of 3? Give a proof or a
counterexample.