Introduction to Abstract Algebra: Group Theory

Explore the definition of a group: a nonempty set with a binary operation closed under the operation, associative, with an identity and inverses; learn about abelian groups and commutativity.

Explore examples of groups by testing natural numbers and real numbers under addition and multiplication, including claims that natural numbers lack an identity and real numbers lack inverses for zero.

the cube roots of unity form a group under multiplication, consisting of 1, ω, and ω^2, with identity, closure, and inverses, an abelian group.

Explore the fourth roots of unity and their structure as a group under multiplication, including x^4=1 and the residue classes modulo 5 forming a group under addition.

Explore how the residue class set {1,2,3,4} forms a group under multiplication modulo 5, with all products remaining in the set.

Explore the group structure of the set of all 2x2 nonsingular real matrices under multiplication, confirming closure, associativity, identity, and inverses in abstract algebra and group theory.

Show that in any group, the only idempotent element is the identity. Derive from associativity that if x*x = x, then x is the identity element.

Learn the left and right cancellation laws in a group, showing AB = AC implies B = C and BA = CA implies B = C, via group axioms.

the lecture shows that for any two elements in a group, the equations x = b have a unique solution by applying the group axioms—closure, associativity, and identity.

Explore the order of a group and the order of its elements, distinguishing finite and infinite groups, and identify the least positive integer n with a^n = e.

Explore how the set {1,3,5,7} forms a group under multiplication modulo 8, determine the order of 1 (which is 1) and 3,5,7 (which are 2), and verify closure and identity.

In a group, if a has order n, then a^k = e iff k is a multiple of n, proven by the division algorithm with k = qn + r.

Explore how the order of an element equals the order of its inverse in a group, using the least positive integer m with a^m = e and divisibility reasoning.

Examine the group structure of the residue classes modulo 9 under multiplication, focusing on the subset {1,2,4,5,7}, verifying closure and exploring element orders.

Examine whether a proposed binary operation on a set satisfies the group axioms and has an identity element, and conclude that this example is not a group.

label the elements e, a, b, construct the multiplication table, and show that a three-element group is abelian by applying the identity and proving ab = ba.

Show that a group G with x^2 = e for all x is abelian by using identity, closure, and associativity to prove ab = ba for all a, b.

Discover how group elements combine under closure, identify inverses, and recognize that if every element is its own inverse then the group is abelian.

In a group where every non-identity element has order 2, the group is abelian, as AB = BA for any elements A and B.

This example examines a group that has a unique element of order two, selects such an element, and uses associativity to derive a relation for all group elements.

This example uses definitions to show that a group is abelian if and only if ab = ba for all a, b in G, using associativity and cancellation.

Define a subgroup as a subset of a group that satisfies the same axioms under the same operation, such as z under addition being a subgroup of q under addition.

Prove the subgroup criterion: a nonempty subset of a group is a subgroup iff for all a,b in it, ab^{-1} lies in it, ensuring identity, inverses, and closure.

Show that the intersection of any collection of subgroups is a subgroup by proving closure under the operation, and that it contains the identity and inverses.

In this example, the intersection of subgroups is always a subgroup, but the union of subgroups may not be; 3 times 5 ≡ 7 (mod 8) is outside the union.

Explore when the union of two subgroups forms a subgroup. Show that union is a subgroup iff one subgroup contains the other, and note the intersection is always a subgroup.

Demonstrate that the union of two subgroups is a subgroup if and only if one is contained in the other, using a contradiction argument and closure properties.

this example examines a mod 4 group under addition, identifying the subgroups {0}, {0,2}, and the entire group, and explains why {0,1} and {0,3} fail to be subgroups.

Explore the Klein four group by examining elements E, A, B, and C and constructing its multiplication table from the identity. Discuss the subgroups, including the trivial and whole group.

Prove that C minus zero is a group under multiplication by detailing closure and inverses, and show that elements on the unit circle form a subgroup.

Defines cyclic groups as groups where every element is a power of a single generator, and explains that some groups, like the additive rationals, are not cyclic.

Let G be a cyclic group generated by a and H a subgroup; choose the least positive m with a^m in H, then H = <a^m>, so H is cyclic.

Every cyclic group is abelian, proven by showing that elements in a group generated by a single element commute under the operation.

Explore the quaternion group Q8, its eight elements i, j, k, and their multiplication rules, proving it is non-abelian and not cyclic through the multiplication table.

Construct the Cayley multiplication table for a cyclic group, identify the identity and its powers a, a^2, a^3, a^4, a^5, and prove closure and the cyclic structure.

In this example, the group with elements e, a, b, c is shown to be abelian but not cyclic, by examining element types and the product table.

Prove that a cyclic group of order n has exactly one subgroup of order d for every divisor d of n, via existence and uniqueness arguments using Lagrange's theorem.

Show that in an infinite cyclic group generated by a, every element is a power a^n, and that the only generators are a and a^{-1}, proving uniqueness.

Examine how to determine element orders in a cyclic group of order 24, including the identity, by finding the least positive m with a^m = e.

Find all subgroups of a cyclic group of order 12 by using divisors of 12, identifying subgroups of orders 1, 2, 3, 4, 6, and 12, with corresponding generators.

Define cosets in a group using a subgroup: learn left cosets gH and right cosets Hg and examine the elements forming these sets.

The group of residue classes modulo six has the subgroup {0,2,4} with two left cosets {0,2,4} and {1,3,5}, illustrating cosets and residue classes.

An exploration of a group on elements 1, 3, 5, 7 under multiplication, identifying subgroups and both left and right cosets, and noting the identity element.

Showcases how the collection of left or right cosets of a subgroup of G partitions G, with union equal to G and intersections empty.

This lecture proves a one-to-one correspondence between left and right cosets in a group by defining a well-defined bijection between the cosets when the subgroup is normal.

Prove that the order of a finite subgroup divides the order of the group by defining the index, partitioning G into left cosets, and counting via the coset partition.

Using Lagrange's theorem, a group of order 89 cannot have subgroups of orders 12, 16, or 24 because these orders do not divide 89.

Demonstrate that a group of order 47, a prime, has no proper subgroups; the only subgroups are the identity and the group itself.

Examine the cyclic group of order 60, identify all subgroups corresponding to the divisors of 60, and show subgroups generated by a^k for each divisor of 60.

Identify all subgroups of the cyclic group generated by a, whose order is 18, using the previous example as a guide.

The set {a + b sqrt(2): a, b ∈ rationals, not both zero} forms a subgroup of the nonzero real numbers under multiplication, with closure and inverses ensured.

examine the set of complex numbers a + b i with not both zero under multiplication and prove the intersection of finite coprime-order subgroups is the identity.

Define permutations as functions from a set to itself, emphasizing that a permutation is one-to-one and onto. Learn the defining properties that characterize these rearrangements.

Show that every permutation in group theory can be expressed as a product of transpositions and as a product of cycles, with illustrated mappings.

Learn how to multiply permutations by composing two permutations, tracing each element’s image through the first and second permutations, with worked examples.

Learn to find the inverse of a permutation and decompose it into disjoint cycles, identifying cycle structures and examples of the cycles involved.

Identify permutation cycles, convert each cycle to transpositions, and determine that the permutations are even; compute order as the least common multiple of cycle lengths.

Determine whether a given permutation is even or odd by analyzing its cycle decomposition and counting transpositions. Explore how cycle structure and transpositions reveal whether the permutation is even or odd.

Explore the S3 symmetric group as the motion group of a triangle, listing identity, transpositions, and 3-cycles, and show how rotations and translations generate all six elements.

Examine the D4 dihedral group of motion, covering 90° and 270° rotations and translations along axes and diagonals, and construct its multiplication table for elements like E, a, b, ab.

Explore the dihedral group D5, the pentagon’s symmetry, detailing its ten elements E, A, A^2, A^3, A^4, B, AB, A^2B, A^3B, A^4B, and how rotations and reflections generate all symmetries.

Examine how a function like f(x)=ln x serves as a homomorphism between the multiplicative and additive group structures, and define monomorphisms, epimorphisms, and isomorphisms in this context.

Define a complex as a subset of a group, illustrated by s3 elements. Analyze when two complexes commute and show immutability in the xy versus yx relationship.

Define the normalizer of a subset X in a group G as the set of elements that commute with X, and illustrate with the Klein four group X = AB.

Explore the normalizer of an element x in the cyclic group c6 through concrete checks of its elements, showing the normalizer equals c6.

Examine the normalizer of x in S3 by testing conjugates with the identity, a, b, and their powers, noting when elements remain within the normalizer in S3.

Identify the identity element as the only normalizing element in S3 in this example, illustrating the concept clearly.

Show that the normalizer of X in G is a subgroup by proving closure under multiplication and inverses, using that B in N_G(X) implies B X B^{-1} = X.