EXAM
Exam 1
Math 3360–001, Fall 2013
Oct. 23, 2013
• Write all of your answers on separate sheets of paper.
You can keep the exam questions when you leave.
You may leave when finished.
• You must show enough work to justify your answers.
Unless otherwise instructed,
give exact answers, not
√
approximations (e.g., 2, not 1.414).
• This exam has 5 problems. There are 310 points
total.
Good luck!
100 pts.
Problem 1. In each part, give the definition of the indicated term or concept,
or state the indicated theorem.
A. A group.
B. A subgroup of a group G.
C. A normal subgroup H of a group G.
D. The quotient group G/H, where H is a normal subgroup of G.
E. A group homomorphism.
90 pts.
Problem 2.
A. Calculate the following product of permuations in S5 .
1 2 3 4 5
1 2 3 4 5
5 3 4 1 2
3 5 4 2 1
B. Consider the permutation in S7
1 2
σ=
5 6
3
2
4
3
5
1
6
4
7
7
i. Draw the directed graph for σ.
ii. What is the sign of σ?
iii. Write σ as a product of disjoint cycles.
C. Consider the product cycles in S5
τ = (1, 3, 4, 2)(4, 5, 1, 3).
These cycles are not disjoint. Rewrite τ as the product of disjoint cycles.
40 pts.
Problem 3. Let H = h4i be the subgroup of Z12 generated by 4
A. Write out the elements of H.
B. Write out the elements of all the distinct cosets of H in Z12 and determine
which of the cosets
a + H,
a ∈ Z12
are equal to each other. How many elements are there in Z12 /H?
1
40 pts.
Problem 4. Let X be a set and let ∼ be an equivalence relation on X. Recall
that
[x] = {y ∈ X | x ∼ y}
is the equivalence class of an element x ∈ X.
Show that the following conditions are equivlent.
(a) x ∼ y.
(b) [x] = [y].
(c) [x] ∩ [y] 6= ∅.
40 pts.
Problem 5. Let G and H be groups and let ϕ : G → H be a group homomorphism. Recall that
ker(ϕ) = {g ∈ G | ϕ(g) = e}.
A. Show that ker(ϕ) is a subgroup of G.
B. Show that ker(ϕ) is a normal subgroup of G.
2