Courses / Mathematics / Abstract algebra I: groups and symmetry Mathematics · unit version 1 Abstract algebra I: groups and symmetry A proof-based group-theory unit with finite computations, structural theorems, counterexamples and complex representation theory through character completeness, semidirect products, small-order classification and geometric rotation groups. Rings, fields and Galois theory belong to later units. Submitted writing requires mathematical review; these pages alone do not establish semester-course equivalence.
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For p=(1 2) and q=(2 3), apply q first and then p to input 1. What if you reverse them? Group operations can be noncommutative, so composition order must be explicit.
Compare reasoning and find a refresher p(q(1))=p(1)=2, whereas q(p(1))=q(2)=3. This one input proves the two composed functions differ.
Review function composition
Find the inverse of 3 modulo 7, and explain why 2 has no multiplicative inverse modulo 6. Residue operations supply concrete examples and counterexamples for group axioms.
Compare reasoning and find a refresher 3·5=15≡1 mod7, so the inverse is5. Every 2k is even and cannot be congruent to1 modulo6; equivalently gcd(2,6)≠1.
Review modular inverses
What must be checked to define a function on equivalence classes using a representative? Quotient groups require a result independent of how a class is named.
Compare reasoning and find a refresher If two representatives belong to the same class, the proposed rule must give the same output class. A rule that changes under replacement is not well-defined on the quotient.
Review equivalence relations
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Group theory: compute, construct and prove structure
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Cumulative abstract-algebra practice examination
Ten original multipart questions connect this unit's methods. Write separately, then review worked arguments and partial-credit criteria. This public practice form does not save responses or award mastery.
Lesson sequence Each linked lesson opens on its own page. Expand its details here when you want to check the objectives before choosing.
Lesson details and objectives Test closure, associativity, identity and inverses without assuming commutativity.
Verify a group from definitions. Disprove a proposed group with a specific failed axiom. Prove uniqueness of identity and inverses. 6 practice tasks.
Lesson details and objectives Use the one-step test and distinguish intersection from union.
Prove the subgroup test. Construct subgroups and counterexamples. Determine generated subgroups. 6 practice tasks.
Lesson details and objectives Classify subgroups of a cyclic group and compute element orders.
Prove the order-of-a-power formula. Determine all subgroups of a finite cyclic group. Distinguish group order from element order. 6 practice tasks.
Lesson details and objectives Use disjoint cycles, inverses and orders.
Convert a permutation between mapping and cycle notation. Compute right-to-left products. Prove the order formula for disjoint cycles. 6 practice tasks.
Lesson details and objectives Define sign and use its kernel.
Prove parity is well-defined. Compute signs from cycles. Identify the alternating subgroup. 6 practice tasks.
Lesson details and objectives Describe polygon symmetries with normal forms.
Derive the dihedral multiplication rule. Distinguish Dₙ from a cyclic group of order 2n. Determine reflection orders and conjugation. 6 practice tasks.
Lesson details and objectives Prove the partition theorem before using divisibility.
Prove cosets partition a group. Derive Lagrange’s theorem for finite groups. Diagnose false converses. 6 practice tasks.
Lesson details and objectives Connect conjugation, normality and quotient operations.
Prove the quotient product is well-defined for a normal subgroup. Exhibit a failed product for a nonnormal subgroup. Compute quotient groups. 6 practice tasks.
Lesson details and objectives Prove kernel and image properties and distinguish injectivity from surjectivity.
Verify a homomorphism on arbitrary elements. Prove the kernel criterion for injectivity. Identify image and fibers. 6 practice tasks.
Lesson details and objectives Prove the first theorem and use the subgroup/quotient correspondences.
Prove the induced quotient map is well-defined and bijective. Apply second and third isomorphism theorems with hypotheses. Use subgroup correspondence. 6 practice tasks.
Lesson details and objectives Compute orders and distinguish products from cyclic groups.
Prove the product construction satisfies the axioms. Determine element orders in finite products. Apply the coprime cyclic-product criterion. 6 practice tasks.
Lesson details and objectives Use primary cyclic factors and uniqueness invariants.
State the finite abelian structure theorem. List types for a specified order using partitions. Distinguish types by element orders and torsion counts. 6 practice tasks.
Lesson details and objectives Prove orbit-stabilizer through a coset bijection.
Verify action axioms. Compute orbits and stabilizers. Distinguish faithful and transitive actions. 6 practice tasks.
Lesson details and objectives Use centralizers to measure conjugacy classes.
Prove conjugation is an action. Derive the class equation. Compute representative classes in a small group. 6 practice tasks.
Lesson details and objectives Prove nontrivial center and classify order p².
Use the class equation for a finite p-group. Prove a cyclic central quotient forces commutativity. Classify groups of order p². 6 practice tasks.
Lesson details and objectives Use existence, conjugacy and number congruences with their hypotheses.
State all three Sylow conclusions. Prove the conjugacy/counting argument through actions. Apply divisibility and congruence jointly. 6 practice tasks.
Lesson details and objectives Classify all groups of order six and distinguish order-eight examples.
Prove the order-six classification. Distinguish nonisomorphic groups with structural invariants. Separate necessary constraints from a complete classification. 6 practice tasks.
Lesson details and objectives Combine quotient, action and counting methods without circular reasoning.
Select and justify a structural method. Repair a proof with a missing hypothesis. Separate computational evidence from a general proof. 6 practice tasks.
Lesson details and objectives Prove derived-series and extension criteria and the A₅ obstruction.
Use commutators to form the universal abelian quotient. Prove subgroup, quotient and extension rules. Distinguish solvable S₃/S₄ from nonsolvable A₅/S₅. 6 practice tasks.
Lesson details and objectives Turn finite-group actions into invariant geometry and prove complete reducibility.
Construct and compare complex representations. Prove averaging, invariant complements and Maschke decomposition. Diagnose a characteristic-p failure. 6 practice tasks.
Lesson details and objectives Prove Schur's lemma and distinguish its field and multiplicity hypotheses.
Solve intertwiner equations on generators. Prove kernel/image and scalar endomorphism results. Compare complex and real irreducibility. 6 practice tasks.
Lesson details and objectives Prove orthogonality, multiplicities and completeness using averaged Hom spaces.
Derive character identities and orthogonality. Prove the regular representation and dimension-square formulas. Establish class-function completeness and column orthogonality. 6 practice tasks.
Lesson details and objectives Connect matrices, irreducible characters and actual invariant subspaces.
Prove standard S₃ irreducibility and exhaust its character table. Decompose ordered-pair actions by two methods. Construct explicit irreducible summands and reject false characters. 6 practice tasks.
Lesson details and objectives Construct and verify external and internal semidirect products.
Identify automorphisms of cyclic groups. Prove the semidirect product axioms. Recover an action from an internal decomposition. 6 practice tasks.
Lesson details and objectives Use complements and actions to classify groups of order pq.
Prove the section and complement criteria. Distinguish cyclic quotients from split extensions. Prove existence and uniqueness in the order-pq classification. 6 practice tasks.
Lesson details and objectives Prove exhaustion, existence and distinction of all five types.
Derive the nonabelian normal forms. Construct dihedral and quaternion multiplication consistently. Separate the five types and test extension splitting. 6 practice tasks.
Lesson details and objectives Connect orthogonal matrices, circle rotations and the fixed point of a finite isometry group.
Derive O(2) and its semidirect product structure. Classify finite plane orthogonal subgroups. Prove the orbit-average fixed-point argument. 6 practice tasks.
Lesson details and objectives Derive axis-angle rotations and the necessary finite-group pole data.
Prove the fixed-axis and Rodrigues formulas. Count pole stabilizers and orbits. Distinguish necessary orbit data from a completed classification. 6 practice tasks.
Lesson details and objectives Construct SU(2), prove its rotation homomorphism and track the two possible lifts.
Build associative quaternion arithmetic and its norm. Prove the rotation map, kernel and surjectivity. Compute half-angle lifts and their ambiguity. 6 practice tasks.
Lesson details and objectives Turn a two-point orbit into a complete geometric classification.
Prove the antipodal orbit and invariant-axis argument. Construct the exact rotation and half-turn normal forms. Handle the order-four boundary without a preferred axis. 6 practice tasks.
Lesson details and objectives Verify solid geometry, supporting faces and symmetry orders from exact coordinates.
Certify the tetrahedral, octahedral and icosahedral models. Prove their rotation-group orders. Distinguish a construction from a classification theorem. 6 practice tasks.
Lesson details and objectives Use convex hulls and forced faces to finish the exceptional conjugacy cases.
Prove the edge bound and triangular-face consequence. Determine edge lengths from vertex stabilizers. Prove geometric uniqueness by adjacent-face propagation. 6 practice tasks.
Lesson details and objectives Prove faithful actions and recover A4, S4 and A5 without identifying groups by order alone.
Identify tetrahedral and octahedral actions and their kernels. Derive icosahedral conjugacy classes geometrically. Prove the five-coset action identifies the icosahedral group with A5. 6 practice tasks.
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