Abstract Algebra: Group Theory with the Math Sorcerer

Explore the additive group of integers modulo n. Remainders 0 to n-1 are the elements; 0 is the identity.

Define a group on the power set of x with the symmetric difference. The empty set is the identity, every subset is its own inverse, and the operation is associative.

Prove that the set of all nonzero diagonal 2x2 matrices forms a group under matrix multiplication by verifying closure, associativity, identity, and inverses.

In a group, a one-sided inverse implies a two-sided inverse, showing that if X Y = E or Y X = E, then Y is the inverse of X.

Show in a group that the inverse of the product xy equals y^{-1} x^{-1} by careful use of inverses, associativity, and the identity.

Explore how the order of an element equals the order of its inverse, with case analysis for finite and infinite order, and a careful proof strategy.

Assume a generator exists for the real numbers under addition and derive a contradiction using 1/2 x not belonging to the set of all integral multiples.

Prove that every cyclic group is abelian by taking a generator and expressing elements as powers, then use commutativity of exponent addition to show ab = ba.

Define subgroups as subsets that are groups under the same operation, with examples like integers under addition, cyclic subgroups, the center of a group, and SL(n, C) inside GL(n, C).

Show how a finite, non-empty subset H of a group G, closed under the group operation, must be closed under inverses, making H a subgroup.

Define a function and its notation, identify domain and codomain, and classify mappings as one-to-one, onto, or bijective with examples such as sine, inverse functions, and logarithms.

Prove that the group map F(X) = X G^{-1} is a bijection by using cancellation to show one-to-one and inverses to establish onto.

Explore the symmetric group on a set x, whose elements are bijections and whose operation is function composition. Learn about identity, inverses, and the array notation for composing permutations.

Learn how to convert a permutation in the symmetric group of degree five into cycle notation, using two-line or array notation with explicit cycles.

Multiply two permutations in degree six using array notation and right-to-left function composition; the lecture walks through mapping each element from 1 to 6 to produce the result.

Multiply two cycles in the symmetric group of degree four by right-to-left composition, tracing the mappings to yield the permutation (1 2) with 3 fixed.

Explains relations as subsets of a Cartesian product, defines reflexive, symmetric, and transitive properties, and shows how these yield equivalence relations with examples like same parents and modular congruence.

Show that in a group with subgroup H, the equivalence class of g equals the right coset Hg. State that two cosets are equal iff a b^{-1} ∈ H.