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Nigeria's free digital learning hub. Videos, lecture notes, practice questions and leaderboards aligned with CCMAS, NBTE, NCCE, WAEC, NECO, IJMB and JAMB curricula.
Mathematics · 300 Level
This is the complete lecture note for MTH 306 Abstract Algebra II, prepared for Mathematics students at 300 Level. You can read it online below, open it in the full viewer, or download the PDF for offline study. It pairs with the course's video lectures, so you can read and watch together.
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Series 1 Normal Subgroups, Quotient Groups, and Isomorphism Theorems • Part 1.1: Normal Subgroups • Definition and properties of normal subgroups • Examples of normal subgroups in familiar groups • Criteria for normality • Part 1.2: Quotient Groups • Definition of quotient groups • Construction of quotient groups using normal subgroups • Properties and examples of quotient groups • Part 1.3: Isomorphism Theorems • First isomorphism theorem • Second and third isomorphism theorems • Applications of isomorphism theorems • Part 1.4: Monomorphisms and Group Homomorphisms (supporting concepts) • Definition of homomorphisms and monomorphisms • Kernel and image of a homomorphism • Relationship with normal subgroups and quotient groups Introduction Groups are central objects in abstract algebra, and understanding their internal structure is essential for deeper study. One of the most important aspects of group theory is the concept of normal subgroups, which provide the foundation for constructing new groups called quotient groups. These ideas are not only theoretical but also practical, as they allow us to simplify complex groups and reveal hidden relationships. This Series will help you appreciate how these constructions fit into the broader framework of algebra and why they are indispensable for advanced study.
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS The aim of this Series is to equip you with the ability to define and explain normal subgroups, construct quotient groups, and apply the isomorphism theorems to analyse group structures. We shall commence this Series by examining the definition and properties of normal subgroups, supported with examples and criteria for normality. Next, we will explore quotient groups, focusing on how they are constructed from normal subgroups and their key properties. Following that, we will study the isomorphism theorems, beginning with the first theorem and then moving on to the second and third, together with their applications. Finally, we will round off the Series by considering monomorphisms and group homomorphisms, highlighting their kernels and images, and showing how they connect back to normal subgroups and quotient groups. Series Learning Outcomes Upon completing this Series, you should be able to: • SLO 1.1 define normal subgroups and explain their properties, • SLO 1.2 construct quotient groups using normal subgroups and illustrate with examples, • SLO 1.3 state and apply the first, second, and third isomorphism theorems, • SLO 1.4 describe monomorphisms and group homomorphisms, and analyse their kernels and images in relation to normal subgroups and quotient groups. 1.1 Normal Subgroups Normal subgroups are a fundamental concept in group theory. They allow us to construct new groups, called quotient groups, and play a central role in the isomorphism theorems. Understanding normal subgroups is therefore essential for progressing in abstract algebra. 1.1.1 Definition of normal subgroups A normal subgroup of a group Gis a subgroup Nsuch that for every element g∈G, the left coset gNequals the right coset Ng. In other words, Nis invariant under conjugation by elements of G. Formally, N⊴Gif and only if gNg-1⊆Nfor all g∈G.
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Fill in the blank: A subgroup Nof Gis normal if for every g∈G, the left coset gNequals the __________ coset. 1.1.2 Properties of normal subgroups • Every subgroup of an abelian group is normal. • The trivial subgroup eand the whole group Gare always normal. • Normal subgroups are precisely the kernels of group homomorphisms. Properties of normal subgroups • Subgroups of abelian groups are normal • Trivial and whole group are normal • Kernels of homomorphisms are normal Fill in the blank: Normal subgroups are precisely the __________ of group homomorphisms. 1.1.3 Examples of normal subgroups • In the group of integers Z,+, every subgroup nZis normal. • In the symmetric group S3, the alternating group A3is a normal subgroup. • In non-abelian groups, not all subgroups are normal, which makes identifying them important.
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Examples of normal subgroups Group Subgroup Normal? Z 2Z Yes S3 A3 Yes S3 e,(12) No Fill in the blank: In the symmetric group S3, the alternating group A3is a __________ subgroup. Pilot questions 1.1 1. A subgroup Nof Gis normal if: A. gN=Ngfor all g∈GB. gNNgfor some g∈GC. Nis abelian D. None of the above (SLO 1.1) 2. Which of the following is always a normal subgroup? A. The trivial subgroup eB. The whole group GC. Both A and B D. None of the above (SLO 1.1) 3. Normal subgroups are precisely: A. Kernels of group homomorphisms B. Images of group homomorphisms C. Centres of groups D. None of the above (SLO 1.1) 4. In the group of integers Z,+, the subgroup 2Zis: A. Normal B. Not normal C. Trivial D. None of the above (SLO 1.1) 5. In the symmetric group S3, the subgroup e,(12)is: A. Normal B. Not normal C. Trivial D. None of the above (SLO 1.1) 1.2 Quotient Groups Quotient groups are constructed from normal subgroups and provide a way of simplifying the structure of a group. They allow us to study groups by breaking them down into smaller, more manageable pieces. 1.2.1 Definition of quotient groups A quotient group is formed when a group Gis partitioned into cosets of a normal subgroup N. The set of cosets G/N={gN:g∈G}becomes a group under the operation (gN)(hN)=(gh)N. The requirement that Nis normal ensures that this operation is well defined.
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Fill in the blank: A quotient group is formed when a group Gis partitioned into cosets of a __________ subgroup. 1.2.2 Construction of quotient groups To construct a quotient group: 1. Identify a normal subgroup Nof G. 2. Form the set of cosets G/N. 3. Define the group operation on cosets: (gN)(hN)=(gh)N. Diagram showing cosets forming a quotient group 1.2.3 Properties and examples of quotient groups • The identity element of G/Nis Nitself. • The inverse of gNis g-1N. • Quotient groups often simplify the study of group structures by reducing complexity.
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Examples: • In Z, the subgroup nZis normal, and the quotient group Z/nZconsists of integers modulo n. • In S3, the subgroup A3is normal, and the quotient group S3/A3is isomorphic to Z2. Examples of quotient groups Group Normal Subgroup Quotient Group Z nZ Z/nZ S3 A3 Z2 Fill in the blank: The quotient group Z/nZconsists of integers taken modulo __________. Pilot questions 1.2 1. A quotient group is formed when: A. A group is partitioned into cosets of a normal subgroup B. A group is partitioned into cosets of any subgroup C. A group is partitioned into elements only D. None of the above (SLO 1.2) 2. The operation on cosets in a quotient group is: A. (gN)(hN)=(gh)NB. (gN)(hN)=(g+h)NC. (gN)(hN)=(g-h)ND. None of the above (SLO 1.2) 3. The identity element of a quotient group G/Nis: A. Nitself B. eC. gNfor some gD. None of the above (SLO 1.2) 4. The quotient group Z/nZconsists of: A. Integers modulo nB. Rational numbers C. Real numbers D. None of the above (SLO 1.2) 5. In S3, the quotient group S3/A3is isomorphic to: A. Z2B. Z3C. Z6D. None of the above (SLO 1.2) 1.3 Isomorphism Theorems The isomorphism theorems are powerful results in group theory that describe the relationships between groups, subgroups, and quotient groups. They provide a systematic way of understanding how different group structures are connected. 1.3.1 The first isomorphism theorem The first isomorphism theorem states that if ϕ:G→His a group homomorphism with kernel K, then the quotient group G/Kis isomorphic to the image of .
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Formally: G/K≅Im(ϕ) This theorem shows that every homomorphism naturally gives rise to a quotient group. 1.3.2 The second isomorphism theorem The second isomorphism theorem states that if Ais a subgroup of Gand Bis a normal subgroup of G, then: AB/B≅A/(A∩B) This theorem relates the product of subgroups to their intersection and quotient structures. 1.3.3 The third isomorphism theorem The third isomorphism theorem states that if Nand Hare normal subgroups of Gwith N⊆H, then: (G/N)/(H/N)≅G/H This theorem describes how successive quotients relate to one another. Summary of isomorphism theorems • First: G/K≅Im(ϕ) • Second: AB/B≅A/(A∩B) • Third: (G/N)/(H/N)≅G/H Theorem Focus Key Insight First Homomorphisms A group modulo a kernel is its image. Second Subgroup Intersections Relates products and intersections of subgroups.
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