Abstract Algebra: Group Theory with the Math Sorcerer

Defines a group as a nonempty set with an associative binary operation, an identity element, and inverses, with examples from integers under addition and nonzero rationals under multiplication.

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

Examine the Klein four group, a four-element abelian group with identity E and elements A, B, C, where A^2=B^2=C^2=E and AB=C, BC=A, CA=B; its cyclic subgroups show it is noncyclic.

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.

The set of real numbers excluding one forms a group under the operation a star b = a + b − ab. It proves associativity, identity (0), and inverse (−a/(1−a)).

Show that in a group, every element has a unique inverse by proving any two inverses must be equal, using associativity and the identity.

Explain a short, clever proof that the inverse of the inverse equals the original element in a group, using the identity element to connect G and G inverse.

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.

The video proves cancellation laws in a group: if ab = ac, then b = c, and if ac = bc, then a = b, using identity, inverses, and associativity.

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.

Define the order of an element as the smallest exponent n with g^n = e; finite or infinite. Q*, -1 has order 2; in Z21, 6 has order 7.

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.

Show that if g^n = e, the order m of g divides n by writing n = m q + r and noting g^r = e forces r = 0.

Explore cyclic groups, generators, and the notation for powers of an element X, with examples from integers modulo n and the real numbers to illustrate when a group is cyclic.

Prove that if G is cyclic generated by X, then G is generated by X inverse. Show that every element g in G equals X^n, and g inverse equals (X^{-1})^n.

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).

Demonstrates that the set of multiples of two forms a subgroup of the integers under addition by verifying nonempty, closure under addition, and closure under inverses.

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.

Demonstrate that the intersection of subgroups H and K of G is a subgroup by verifying non-empty, closure under the group operation, and closure under inverses.

Explore why the center of a group contains elements that commute with every element. Prove the center is a subgroup by showing nonemptiness, closure under multiplication, and closure under inverses.

Prove that H = { x in G | x^2 = e } is a subgroup of G by showing non-emptiness, closure under multiplication, and inverses, using abelian property.

Show that SL(n,R) is a subgroup of GL(n,R) by verifying nonempty, closure under multiplication, and inverses, using det(AB)=det A det B and det(A^{-1})=det(A)^{-1}.

Demonstrate that the centralisers of G form a subgroup by verifying non-emptiness, closure under the group operation, and closure under inverses.

Explore when the direct product of finite cyclic groups is cyclic by requiring pairwise relatively prime orders, using the greatest common divisor to test across multiple factors.

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.

Show the determinant map F from GL(2, C) to the nonzero complex numbers is surjective. For any nonzero z, choose A = [z 0; 0 1] so that det(A) = z.

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.

Convert a permutation in the symmetric group of degree eight to array (two-line) notation, revealing the mapping 1→4, 4→2, 2→3, 3→5, 5→7, 7→6, 6→1, with eight fixed.

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 permutations in S5 by composing a 4-cycle and a 3-cycle right-to-left, revealing the resulting mappings and closed cycles.

Multiply two cycles in the symmetric group S6 using right-to-left composition, identify fixed points, and compute the product to obtain the permutation with five fixed.

Multiply two cycles in the symmetric group S7 from right to left to compute the product. The result is (1 4)(2 6)(5 7), with 3 fixed.

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.

Equivalence classes under a relation are either identical or disjoint, non-empty by reflexivity, and the collection partitions the set with a union that covers it.

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.