Download problem set #1

Problem Set #1
Due: 12 September 2007
1. Let X be a set. The power set P ( X ) is the set of all subsets of X. For A, B ∈ P ( X ),
the symmetric difference is the set A 4 B := ( A \ B) ∪ ( B \ A). Prove that P ( X )
equipped with the symmetric difference operation is an abelian group.
2. Let [10] := {1, 2, . . . , 10}. Define α : [10] → [10] by
α( j) = the remainder after dividing j8 + 9j3 by 11 .
(a)
(b)
(c)
(d)
Show α is a permutation.
Find the cycle decomposition of α.
Compute the inverse of α.
Express α as a product of transpositions.
3. This problem gives a shorter list of group axioms. Let H be a set equipped with an
associative binary operation ∗ satisfying the following two properties:
left surjectivity: left multiplication by any element gives a surjective map; in
other words, for each g ∈ H, the left translation τg : H → H defined by
τg (h) = g ∗ h is surjective.
right surjectivity: right multiplication by any element gives a surjective map; in
other words, for each g ∈ H, the right translation µ g : H → H defined by
µ g (h) = h ∗ g is surjective.
(a) Show that some element in H has a left identity (which a priori may depend
on the element).
(b) Show that the left identity for some element in H is actually a left identity for
all elements in H.
Hint: Use left surjectivity and associativity.
(c) Show that H has a right identity.
(d) Show that the left identity and right identity are equal, so H simply has an
identity.
(e) Show that each element in H has a left inverse and a right inverse.
(f) Show that the left inverse equals the right inverse, so each element in H simply
has an inverse.
(g) Prove that H is a group.
4. Use MathSciNet: www.ams.org/mathscinet/ (from a Queen’s computer) and
the ArXiv: arxiv.org (or front.math.ucdavis.edu) to answer the following
questions:
(a) Estimate the number of publications that have the word “permutation” in their
title.
(b) How many “Commutative Algebra” e-prints were added to the archives in
July 2008?
(c) What was the most unusual paper you found?
MATH 893: page 1 of 1