Homework 5
Due Thursday, May 18, at the beginning of discussion
1. An automorphism of a group G is a group isomorphism ? : G ! G. List the automor- phisms Z ! Z and prove that these are the only automorphisms.
2. Remember to prove your answers in this problem:
(a) Compute the lattice of subgroups of the group Z/45Z and draw the lattice of sub- groups.
(b) Compute the lattice of subgroups of Z/2nZ? 3. We know that every subgroup of Z is cyclic. Find the generators of the following sub-
groups and prove your answers.
(a) aZ + bZ (b) aZ bZ
4. Let (G, ?) be a group. Fix an element a in G. The centralizer of a in G consists of all
the elements of G that commute with a. Namely,
C(a) = {g 2 G: g ? a = a ? g}.
(the set of all elements x in G that commute with a).
(a) Compute the centralizer of 3 in (Z, +). (b) Compute the centralizer of the element r 2 D6; and the centralizer of the element
r 2 D8. (c) Show that for every group G and every element a in G, C(a) is a subgroup of G.
(d) In general, which one is bigger, Z(G) and C(a)? Is one a subgroup of the other?
5. Consider the set H = {? 2 S5 : ?(1) = 1 and ?(3) = 3}. (Recall that S5 consists of bijective functions from {1, 2, 3, 4, 5} to {1, 2, 3, 4, 5}.) (a) Show that H is a subgroup of S5.
(b) Show that H is isomorphic to S3.
6. Write down a an isomorphism from Z/2Z ? Z/2Z to a subgroup of S4. You dont need to prove that the map you wrote is an isomorphism.
7. Let ? =
1 2 3 4 5 6 7 8
2 3 4 5 1 7 8 6
! , ? =
1 2 3 4 5 6 7 8
1 3 8 7 6 5 2 4
! .
(a) Compute ??1, ??
(b) Compute ?, ? and ?? as product of disjoint cycles
(c) Compute ? and ? as product of 2-cycles.
8. Find the order of the following permutations:
(a) (145)(2345) 2 S5. (b) (154)(254)(1234) 2 S5. (c) (1574)(324)(3256) 2 S7.
1
(d) ? : Z ! Z where ?(n) = n + 3, as an element of SZ, the group of bijective maps Z ! Z with composition as the operation.
9. What is the maximum order of an element in each of the groups S4, S5, S6, S7, S8? Exhibit
such an element in each case.
Recent Comments