Download MAT 301 Groups and Symmetry - Problem Set 1 Due Wednesday

MAT 301 Groups and Symmetry - Problem Set 1
Due Wednesday, January 27 at 6:10pm
NOTE:
• Students are expected to write up solutions independently. If several students hand in
solutions which are so similar that one or more must have copied from someone else’s
solution, this will be treated as academic misconduct and reported to the department
administration.
• Not all of the questions will be marked. Students will not know in advance which
questions will be marked and are advised to hand in solutions to all of the questions.
• In order to receive full marks for computational questions, all details of the
computation should be included in the solution. Even if the correct final answer is
given, marks will be deducted if some details are left out.
• In solving questions involving proofs, unless the question specifically lists which results
may or may not be used, it is not necessary to reprove facts that have been proved in
class, in the sections of the text that have been covered, or on previous problem sets.
1. Show that the set G = {x ∈ R, 0 ≤ x < 1} is a group with respect to the operation
x ∗ y = x + y − [x + y], where x, y are any two elements of G, and [z] denotes the
biggest integer less or equal than z.
2. How many elements σ such that σ = σ −1 are there in the permutation group S4 ?
3. Let X be a (possibly infinite) set containing at least 3 elements. Show that the group
of all invertible maps of X to itself is not Abelian.
4. Explicitly listing all possible Cayley tables, or otherwise, show that any group of
order 4 is isomorphic either to Z4 , or to Z∗8 .
5. (a) Let V4 be the set of permutations σ ∈ S4 which have the following property:
either σ = e, or there exist distinct i, j, k, l ∈ {1, 2, 3, 4} such that σ(i) = j,
σ(j) = i, σ(k) = l, and σ(l) = k. Show that V4 is a group with respect to
composition of permutations (hint: list all permutations in V4 ).
(b) Is V4 isomorphic to Z4 ? Is it isomorphic to Z∗8 ?
6. Is there n such that Z∗16 is isomorphic to Zn ?
7. Prove that if m ∈ Zn has an inverse with respect to multiplication modulo n, then m
is coprime with n (the converse statement has been proved in class).