Some basic but fun exercises

Here is a list of some fun exercises (at least for me). Many are my own creation, others are inspired from textbooks or from talks. They are graded 1 to 5 stars indicating increasing difficulty.

The math below is rendered with KaTeX. I don't understand why people use the slow and bulky MathJax.

Algebra

• (1* ) Let \( p \) be a polynomial in a single variable. Show that \( x - a \) is a factor of \( p(x) - p(a) \). (Hint: consider \( x^n - a^n \).) Hence show that \(p \) has a root at \( a \) iff \( p \) it has \( x - a \) as a factor.
• (1* ) Show that \( a^p = (a - 1)^p + 1 \) for \( a \in (Z/p)^\times \). Which theorem of Fermat does this prove? (Hint: it's a small one.)
• Let \( G \) be a finite group and define \( \rho(G) = | (x,y) : xy = yx | / |G \times G| \). (This fraction can be thought of as a description of how abelian \( G \) is.)
  • (2** ) Show that \( \rho(G) \leq \frac{1}{2} + \frac{1}{2}|Z(G)|/|G| \)
  • (1* ) Show that if \( G/Z(G) \) is cyclic, then \( G \) is abelian.
  • (1* ) Conclude that if \( \rho(G) > 5/8, \) then \( G \) is abelian.
• (3*** ) Show using Noether's Third Isomorphism Theorem that there is a group isomorphism \( U(2)/U(1) \cong SU(2)/\{\pm I \} \). Then show via the action of \( SU(n) \) on complex space \( C^n \) that the cosets \( SU(n)/SU(n-1) \) can be identified with points in \( S^{2n - 1} \) (i.e. there is a sensible bijection between the two). Hence show there \( U(2)/U(1) \) can be identified with \( RP^3 \).

Analysis

• (3 ***) Let \( f : R \rightarrow R \) satisfy \( f(x + y) = f(x) + f(y) \). Is it linear or continuous?

Combinatorics

• (1* ) Show that if \( (a, b) \) is a solution to \( 2x^2 + 1 = y^2 \), then \( (2ab, 2b^2 - 1) \) is too. Hence show that \( { n \choose 2 } = m^2 \) has infinitely many solutions.
• (2** ) Show the \( n \)-th row of Pascal's triangle consists entirely of odd numbers precisely when \( n = 2^k - 1 \) for some \( k \). (Assume the top row to be the 0-th row.)
• (2**) Compute the number of order-preserving functions (i.e. non-decreasing functions) between \( \{ 0, 1, \ldots, n \} \) and \( \{ 0, 1, \ldots, m \} \).
• (3**) Characterize all positive integers which can be expressed as a sum of at least two consecutive positive integers. (Hint: one direction can be proved by construction. For the other, consider a common formula often used to illustrate proofs by induction.)
• (2**) A group of people are standing in a circle. You have a bag with fruit which you distribute it the following way: choose someone, and give them a piece of fruit. Move clockwise by 1 person, and give that person a piece of fruit too. (The two people who have received fruit are adjacent). Move clockwise by 2 people, and give that person a piece of fruit. Move clockwise by 3 people, 4 people, etc, continuing the pattern until all fruit has been given out. When is it the case that everyone in the circle receives at least one piece of fruit?

Geometry

• (1*) Show that a cube contains a tetrahedron whose volume is 1/3 that of the cube. (Note: the volume of a tetrahedron with side length \( x \) is \( x^3 \sqrt{2} / 12 \) ).

Topology

• (2 **) Determine all closed oriented covering spaces of the torus.
• (2 **) Construct an explicit 2-sheeted covering map of the Klein bottle by the torus.
• (2 **) Let \( \sim \) be the the equivalence relation on the real line \( R \) defined by \( a \sim b \) if \( a - b \in Z \). Denote the quotient space \( R / \sim \) by \( R / Z \).
  • Using the definition of the topology on a quotient space, show that \( R / Z \) is compact.
  • Show that the Euclidean plane \( R^2 \) is Hausdorff.
  • Show that any continuous function from a compact space to a Hausdorff space maps open sets to open sets.
  • Construct a continuous bijection from \( R / Z \) to \( S^1 \), and then conclude that this is a homeomorphism. (Hint: construct the bijection by defining a map from \( R \) to \( S^1 \).)
  • Adapt the argument above to show that \( R^2 / Z^2 \) is homeomorphic to \( S^1 \times S^1 \).
• (3 ***) Construct an explicit map which shows that \( SO(3) \) is diffeomorphic to \( RP^3\).