Math 366 Assignment 3
Due Wednesday, November 3
1. What is the order
1 2 3
(a)
2 1 5
1 2 3
(b)
7 6 1
1 2 3
(c)
1 2 3
1 2
2. Let α =
2 1
of each of the following permutations?
4 5 6
4 6 3
4 5 6 7
2 3 4 5
4 5
4 5
3 4 5 6
1 2 3
and β =
3 5 4 6
6 1 2
4
4
5
3
6
5
. Compute the following:
(a) α−1
(b) αβ
(c) βα
3. (a) What is the order of S6 (the symmetric group on 6 letters)?
(b) How many elements of S6 are odd permutations?
(c) Is the permutation (1356)(247) even or odd?
4. In S3 , find elements α and β such that |α| = 2, |β| = 2, and |αβ| = 3.
5. Find an isomorphism from the group of integers under addition, (Z, +), to the group of even integers
under addition, (2Z, +). Be sure to show that it is an isomorphism (one-to-one, onto, φ(a + b) =
φ(a) + φ(b).
√
6. Show that the mapping φ(x) = x from the group of positive real numbers with multiplication,
(R+ , ∗), to itself is an isomorphism. (An isomorphism from a group to itself is called an automorphism.)
7. Let H = {0, ±3, ±6, ±9, . . . }. Find all cosets of H in Z.
8. Find the cosets of {1, 11} in U (30).
9. Suppose K is a proper subgroup of H and H is a proper subgroup of G. If |K| = 42 and |G| = 420,
what are the possible orders of H?
NOTE: Much of this is from the 6th edition of Gallian’s Contemporary Abstract Algebra.
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