Math 417: Abstract Algebra Homework 1
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Homework 1
Math 417: Abstract Algebra
(Exercises are taken from Algebra: Abstract and Concrete, Edition 2.6 by Frederick M. Goodman.)
1. Exercise § 1.1.3 Determine the rotational symmetrices of a brick with three unequal sides.
2. Exercise § 1.3.3. Here is another way to list the symmetries of the square.
a. Verify that the four symmetries a,b,c,d that exchange the top and bottom faces of the card are a,ra,r2 a,r3 a, in some order. Thus, a complete list of symmetries is {e,r, r2 , r3 ,a,ra,ra2 , ra3 }.
b. Verify that ar = r −1a = r3 a.
c. Conclude that ark = r −ka for all k ∈ Z.
d. Show that these relations suffice to compute every product.
3. An affine transformation of Rn is a function T : Rn → Rn of the form T(x) = Ax + b, where A ∈ Matn ×n(R) and b ∈ Rn.
a. Show that if T, U : Rn → Rn are affine transformations, so is the composite function U 。T
b. Show that T(x) 
Ax + b is a bijection if and only if A is an invertible matrix. We call such T an invertible affine transformation.
c. Show that if T is an invertible affine transformation, then its inverse function is also an invertible affine transformation.
4. Exercise § 1.5.3.
5. Exercise § 1.5.5. Show that any k-cycle (a1 ··· ak) can be written as a product of (k − 1) 2-cycles. Conclude that any permutation can be written as a product of some number of 2-cycles.
6. Exercise § 1.5.9. Let σn denote the perfect shuffle of a deck of 2n cards. Regard σn as a bijective function of the set {1, 2, . . . , 2n}. Find a formula for σn(j) when 1 ≤ j ≤ n, and another formula for σn(j) when n + 1 ≤ j ≤ 2n. (The ”perfect shuffle” is described in Example 1.5.2.)
2023-12-15