Group theory exercises from Herstein Abstract Algebra (Alex Sanchez):
Let G be an abelian group, and let h₁, h₂ ∈ G be elements. Prove that there exists an element h
∈ G whose order is the least common multiple of the orders of h₁ and h₂.
Let G be an abelian group, and let H₁, and H₂ be subgroups. Prove that there exists a subgroup
of G whose order is the least common multiple of the orders of H₁ and H₂.
Topology (Zilu Li):
If f : X → Y is a quotient map, then for each y ∈ Y the set f⁻¹({y}) is connected. If Y is
connected, then so is X.
If a set is connected, then so is its closure.
In a metric space (X,d), the following are equivalent: (a) X is compact, (b) Every infinite
subset of X has a cluster point, (c) Every sequence in X has a convergent subsequence, (d) X is
complete and totally bounded, (e) X is totally bounded and has the Lebesgue property.
Commutative algebra (Raymond Guo):
An integral domain is a PID if and only if every prime ideal is principal.
Sequences (Zachary Banken, Gregory Baimetov):
Beatty's Theorem (aka Rayleigh's Theorem): see wikipedia.
Uspensky's Theorem: it is not possible to partition the natural numbers using 3 or more Beatty
sequences.
