# 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 $$RP^3$$.