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Abstract algebra: name the operation a map must preserve

A formula on residue classes must be well-defined before it can preserve an operation. Even a correct group isomorphism need not preserve multiplication. Use one small map to separate these claims instead of treating “same structure” as a single test.

Course topic: Abstract Algebra

Study route and exercise: University Reading List editorial project. How we curate · Sources checked:

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Scope and responsibility

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.

Official free reading ↗

Book record and access options →

A Book of Abstract Algebra

Charles C. Pinter · Edition: 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.

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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.

Book record and access options →

Check your preparation

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 six-step study plan

These are your own study marks, not an assessment. They are not sent to this site.

  1. 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.

  2. 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.

  3. 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.

  4. 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.

  5. 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.

  6. 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.

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.

  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.

Answer and interpretation

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

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?

  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.

Answer and interpretation

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.

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.

Sources and edition checks

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These are proposed study activities, not official course assignments or a claim that every book has been read cover to cover.

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