Continuing the read of part 3 of Category Theory Illustrated by Jencel P. on stream.

• Previous reading is 20251016180138
• Group isomorphisms
  • Two groups are isomorphic if given an function f from one group to the other and any two elements a and b from the first group, f(a ∘ b) is the same as f(a) ∘ f(b)
  • In group theory, all isomorphic groups are actually the same group. For example, the group of rotations for a triangle and the group of integers under addition with modulo 3 are both the Z₃ group.
  • The smallest group Z₁ has a single element (there is no Z₀ because groups need to have an identity element, and there are no elements in an empty set).
  • Z₂ is a group with two elements, also commonly known as the boolean group.
  • Groups can be combined in such a way that the new group is the cartesian product of the two groups and the operations of either group only affect the respective element in the new group. This process of combining two groups is known as a group product. For example, the group product of two Z₂ groups is known as the Klein four-group and is a non-cyclic Abelian group.
  • An example of a Klein four-group is the group of symmetries of a non-square rectangle.
  • Klein four-group is non-cyclic because there are two different generators (which are not inverses of each other) that you have to use in order to get all of the elements in the set.
  • The Klein four-group is the smallest non-cyclic group.
  • Group products can result in cyclic groups. For example, two groups that are relatively prime (that is, they share no common divisor other than 1) can.
    • For example: Z₂ × Z₃ ≅ Z₆ (read as Z₂ × Z₃ is isomorphic to Z₆)
    • Detexify
      • Someone suggested using Detexify to find source code for Latex: https://detexify.kirelabs.org/classify.html
        Detexify mentions that the comprehensive reference for Latex symbols is at https://www.ctan.org/tex-archive/info/symbols/comprehensive/
    • Consider the cartesian product (0,0), (1,1), (0,2), (1,0), (0, 1), (1,2)
    • You can generate all of the elements in the cartesian product by applying the +1 or the -1 generator to each element.
    • So, there's only one generator for Z₂ × Z₃.
    • This is also known as the Chinese Remainder theorem.
  • Abelian product groups
    • Product groups are abelian if the groups that form them are also abelian.
    • This is true because the generators for a product group only operate on its own part of the group.
    • The only way to make non-cyclic Abelian groups is by having a product of cyclic groups.
    • Law: All Abelian groups are either cyclic or products of cyclic groups.
  • Dihedral groups
    • Consider a regular polygon, like a triangle, with both rotation and reflection.
    • These two operations with a triangle results in the group known as Dih3, which is not abelian because the operations are no longer commutative (flipping and then rotating gives you a different result than rotation and then flipping).
    • Dihedral group - a group of rotations and reflections of any regular polygon.
      • The book says "any 2D polygon" here, but this is not true according to chat and wikipedia which states that "a dihedral group is the group of symmetries of a regular polygon". ¹
  • Groups/monoids categorically
    • Groups and monoids are specific types of categories. I think this means that group theory is an example of category theory.
    • Instead of thinking about monoids as their objects and operation, we can instead think of monoids as a set of operations. For example, instead of the set of integers and the operation +, we can think of it as a set of functions that add some amount (+0, +1, +2, etc...).
    • Any function that takes a pair of objects, can be turned into a function that takes one object and returns a new function which takes the other object and returns the result. This is also known as currying.
    • Currying is named after Haskell Curry, but it was discovered earlier by Moses Schönfinkel.
    • In programming, currying is achieved by a higher-order function.
  • Cayley's theorem
    • I'm not sure what the author means by "permutations" and I am finding it difficult to understand this section.
  • Symmetric groups
    • I still don't know what the author means by "permutations", but apparently this has something to do with Cayley's theorem.
    • The study of group theory started by examining symmetric groups.
  • Monoids as categories
    • Law of Closure functional composition on any two objects always yields the same object. Monoid and group operations have closure and categories do not.
  • Group/monoid presentations
    • The presentation of a group is the set of generators that defines a given group.
    • Z₃ has three morphisms. For example, for a regular triangle, those morphisms are the identity (0 degrees), 120 degrees, and 240 degrees.
    • Since Z₃ is a finite group, applying the morphism enough times gives you the identity morphism.
    • The Klein four-group has four morphisms. For example, for a non-square rectangle with horizontal flip and vertical flip, there is the identity, horizontal flip, vertical flip, and the horizontal and vertical flip (180 degree rotation).
  • Free monoids
    • Free monoids are monoids which can be upgraded to a monoid for free without needing to define anything else. This does not make sense to me because it sure sounds like a "an X is an X" definition.