# Abstract algebra: name the operation a map must preserve — Study worksheet [Study guides](https://unireadinglist.com/guides/abstract-algebra/) [Course topic: Abstract Algebra](https://unireadinglist.com/subjects/mathematics/#course-abstract-algebra) Sources checked: 2026-11-04 Study route and exercise: University Reading List editorial project. A proof-based route through cyclic groups, homomorphisms, kernels and quotients, with a follow-up that adds ring multiplication. Judson is the selected 2026 annual text, Pinter the Dover second-edition reprint, Carter the selected AMS reprint of a visual group-theory work. Carter supports the group route; a full ring course needs a broader text. ## Choose your starting point - [ ] I want a free textbook from groups to rings Use Judson for a rigorous undergraduate course with computational examples and later rings/fields. - [ ] I need shorter steps and exercise-led proofs Use Pinter when small chapters and carefully sequenced exercises help bridge school algebra and abstract arguments. - [ ] I need to see the group structure Use Carter for Cayley diagrams, quotients and homomorphism pictures; pair it with a conventional text for ring theory. ## Compare the three books ### Abstract Algebra: Theory and Applications Thomas W. Judson · 2026 Free main textbook with a focus on proofs Choose it when: You can work with sets, functions and short proofs and want a course extending beyond groups. Use another route when: You want only visual symmetry or expect computation to replace justification. Selected reading: Groups, cyclic groups, homomorphisms and factor groups; then rings and ring homomorphisms. Limits and trade-offs: The broad course and optional Sage activities give several entry points, but you must distinguish experimental evidence from proof. #### Try before choosing For \[k\]↦\[2k\] from Z/12 to Z/8, test two representatives of the same input class. Check your work: Replacing k by k+12 changes the output by 24, which is zero modulo 8; the formula is well-defined. Your notes: ________________________________ ISBN: 9781944325213 [Official free reading](https://judsonbooks.org/abstract-algebra-theory-and-applications/) [Book record and access options](https://unireadinglist.com/subjects/mathematics/abstract-algebra-theory-and-applications/) ### A Book of Abstract Algebra Charles C. Pinter · 2 · 2010 Short-step introduction Choose it when: You need practice constructing arguments through small exercise sequences. Use another route when: You want a comprehensive graduate reference or a complete public solutions manual. Selected reading: Operations, homomorphisms, normal subgroups/quotients and the later ring material in the selected second edition. Limits and trade-offs: Important results can develop in exercises; skipping them can remove a needed proof rather than just extra practice. #### Try before choosing List the kernel and explain why two inputs have the same image exactly when their difference is in it. Check your work: Kernel is \{0,4,8\}; fibers are its cosets, not arbitrary groups of three inputs. Your notes: ________________________________ ISBN: 978-0-486-47417-5 [Book record and access options](https://unireadinglist.com/subjects/mathematics/a-book-of-abstract-algebra/) ### Visual Group Theory Nathan Carter · 2021 Visual group route Choose it when: A diagram of actions or a Cayley graph helps make a quotient concrete. Use another route when: Your syllabus immediately requires a full treatment of rings, ideals and modules. Selected reading: Subgroups, Products and quotients, and The power of homomorphisms; use the introductory diagrams first. Limits and trade-offs: Visualization builds examples, but a visual group correspondence does not certify multiplication in an additional ring structure. #### Try before choosing Draw the four fibers of the worked map and label their common output residues. Check your work: There are four fibers of size three, mapping to 0,2,4,6; a picture still needs an operation-preservation check. Your notes: ________________________________ ISBN: 9781470464332 [Book record and access options](https://unireadinglist.com/subjects/mathematics/visual-group-theory/) ## Check your preparation ### Can I distinguish an integer from its residue class? Your notes: ________________________________ ### Which operation is being preserved? Your notes: ________________________________ ### What is an isomorphism onto the image? Your notes: ________________________________ ## A six-step study plan - [ ] Define the objects Selected reading: Sets/functions and modular arithmetic. What to do: Write domain, codomain and the addition operations with explicit moduli. Check your work: The input and output use different equivalence relations. Your notes: ________________________________ - [ ] Prove well-definedness Selected reading: Homomorphism material. What to do: Replace k by k+12m and show the output class is unchanged. Check your work: The proof handles every integer m, not just one example. Your notes: ________________________________ - [ ] Find the kernel and image Selected reading: Cyclic groups and kernels. What to do: List all twelve inputs, then give a divisibility argument for the kernel and image. Check your work: Kernel size 3 and image size 4 account for all twelve inputs. Your notes: ________________________________ - [ ] Build the quotient map Selected reading: Factor groups and the isomorphism theorem. What to do: List the four cosets and their outputs, and verify the induced map is additive and bijective. Check your work: The result identifies additive groups; no product claim has been proved. Your notes: ________________________________ - [ ] Change the required structure Selected reading: Rings and the follow-up. What to do: Test multiplication using one explicit pair and investigate an identity in the image subring. Check your work: Multiplicative closure alone does not make the same formula a ring homomorphism. Your notes: ________________________________ - [ ] Write a precise theorem statement Selected reading: Your two solved cases. What to do: Replace “the structures are the same” by one correct isomorphism claim and one failed stronger claim. Check your work: Domain, image, operation and kernel appear in the statement. Your notes: ________________________________ ## Worked exercise: A quotient formula with different input and output moduli Define phi: Z/12→Z/8 by phi(\[k\]12)=\[2k\]8, using addition in both groups. Prove the map is well-defined and additive, find its kernel and image, and describe the induced quotient-group isomorphism. Assumptions: Finite additive residue groups; all arithmetic is exact. Bracket subscripts identify the modulus. No multiplication-preservation assumption is made in the main case. Your notes: ________________________________ ## Change an assumption: Adding ring multiplication changes what must be proved Keep the same sets and formula, but now use the usual ring multiplication modulo 12 and modulo 8. Is phi a ring homomorphism, even without requiring it to preserve 1? Is the even-residue image closed under multiplication, and does that image have a multiplicative identity? Your notes: ________________________________ ## Before moving on - Prove a residue-class formula is well-defined. - Construct and verify an induced quotient-group map. - Give an explicit counterexample to an unjustified ring claim. ## Answer and interpretation ### Can I distinguish an integer from its residue class? \[k\] in Z/12 has many representatives. A rule must give the same class in the codomain for all of them. ### Which operation is being preserved? For additive groups check phi(a+b)=phi(a)+phi(b). For a ring also check products, and state whether preserving the multiplicative identity is required. ### What is an isomorphism onto the image? It is a bijective operation-preserving map after the domain is quotiented by the kernel. Name the category: the main result concerns groups. ### A quotient formula with different input and output moduli 1. If k′=k+12m, then 2k′−2k=24m is divisible by 8, so both representatives give the same output class. Also phi(\[a+b\])=\[2a+2b\]=phi(\[a\])+phi(\[b\]). 2. The kernel solves 2k≡0 mod 8, equivalently k≡0 mod 4. In Z/12 it is \{0,4,8\}. The image is \{0,2,4,6\}, with addition modulo 8. 3. Cosets are \{0,4,8\}, \{1,5,9\}, \{2,6,10\}, \{3,7,11\}, mapping respectively to 0,2,4,6. Each image has one coset, so the induced additive map (Z/12)/ker(phi)→im(phi) is bijective. phi is a well-defined additive homomorphism with kernel \{0,4,8\}, image \{0,2,4,6\}, and an additive quotient-group isomorphism of order 4. Test the result: The orders satisfy 12=3·4. Both \[1\]12 and \[13\]12 map to \[2\]8. The original map is not injective; the induced quotient map is. ### Change an assumption: Adding ring multiplication changes what must be proved 1. Take a=b=\[1\]12. Then phi(ab)=\[2\]8, while phi(a)phi(b)=\[4\]8. Products are not preserved, so phi is not a ring homomorphism even under the convention that allows nonunital maps. 2. The even residues are closed under addition and multiplication modulo 8, so they form a subring under the nonunital convention. Closure is a property of the set, not a proof about phi. 3. If an even residue e acted as an identity there, it would satisfy 2e≡2 mod 8. This requires e≡1 mod 4, which is odd, contradicting membership. Thus the image subring has no multiplicative identity. The additive result survives, but phi is not a ring homomorphism. Its image is multiplicatively closed yet has no identity; the earlier group isomorphism does not establish a ring isomorphism. Test the result: Kernel \{0,4,8\} is also an ideal in Z/12, but that fact does not fix phi’s failed product test. A correct quotient theorem can only be applied after its map hypotheses are established. ## Common mistakes ### Testing only representatives in a table A table helps find the answer; well-definedness still concerns all representatives of a class. ### Forgetting the operation “Homomorphism” and “isomorphism” need a named structure and operation. ### Using closure as map preservation A closed image set does not make the chosen map multiplicative. ## Sources and edition checks - [Judson: official annual editions and free text](https://judsonbooks.org/abstract-algebra-theory-and-applications/) The author distinguishes annual print/PDF revisions from the online course, and separates group theory from later rings/fields. - [Dover: selected second-edition reprint](https://store.doverpublications.com/products/9780486474175) The selected Dover second-edition work uses short chapters and exercise sequences for an introductory algebra route. - [MAA publisher-hosted contents: visual group theory](https://www.jstor.org/stable/10.4169/j.ctt5hh9nr) Publisher-hosted contents support the visual group/quotient/homomorphism route; it is not a complete ring-theory text. 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