Group Homomorphism Tool Calculator

How to use

1. Choose a mode: cyclic or direct product.
2. Enter n and m, and r if using product mode.
3. Provide generator images: k, or a and b.
4. Press Compute to see results above the form.
5. Use CSV or PDF buttons to export the result block.

Formula used

Example data table

Generator-driven maps in finite additive groups

Every homomorphism between finite additive groups is determined by images of generators. In the cyclic case, choosing k fixes f(x)=kx mod m, so the tool converts abstract structure into arithmetic. This supports quick exploration of endomorphisms, quotient behavior, and compatibility with the relation n·1=0 in the domain. Because computations are modular, you can test many candidates quickly, compare different codomain sizes, and see how changing k alters subgroup sizes. This is useful in coursework, proof checking, and building intuition about cyclic group morphisms.

Well-definedness and relation preservation

To descend from integers to Z_n, the map must send n to zero in Z_m. The calculator reports whether (m/gcd(m,k)) divides n, which is equivalent to f(n)=0. For product domains, it verifies m|n·a and m|r·b, ensuring both cyclic relations are respected.

Kernel and image as measurable subgroups

The kernel collects inputs mapped to 0, capturing the “lost information” of the map. The image is the subgroup reached in the codomain. For cyclic maps, |Im|=m/gcd(m,k) and |Ker|=n/|Im| when well-defined. For product maps, |Im|=m/gcd(m,a,b), reflecting the subgroup generated by a and b.

Injective, surjective, and structural consequences

Injectivity is equivalent to a trivial kernel, so the tool flags injective when |Ker|=1. Surjectivity means the image size equals m, which often requires gcd conditions to be 1. These indicators help you classify maps as embeddings, quotients, or automorphisms, and they guide expectations about solution multiplicity for f(x)=t.

Cross-checking with the First Isomorphism Theorem

For finite groups, the First Isomorphism Theorem implies |G|/|Ker(f)|=|Im(f)|. The calculator uses this relation to compute sizes even when listing all elements would be large. When enumeration is feasible, it also previews mappings and explicitly lists kernel and image, helping connect formulas to concrete data. When the map is not well-defined, the tool warns you early, so you avoid invalid constructions. For large parameters, it still provides reliable size outputs, letting you document results without brute-force iteration safely.

FAQs

1) What does “well-defined” mean here?

It means the formula respects the domain’s defining relations. For Z_n→Z_m, you need f(n)=0 in Z_m. In product mode, both n·(1,0) and r·(0,1) must map to 0.

2) Why is f(x)=k·x enough in the cyclic mode?

Z_n is generated by 1 under addition. Any homomorphism is determined by the image of 1, called k. Linearity then forces f(x)=x·k, reduced modulo m.

3) How are image and kernel sizes computed for large inputs?

When listing every element would be heavy, the tool uses gcd-based subgroup formulas and the finite relation |G|/|Ker|=|Im| to compute sizes reliably, without enumerating all pairs.

4) What makes the map injective or surjective?

Injective means only the identity element maps to 0, so |Ker|=1. Surjective means every element of Z_m is hit, so |Im|=m. Both depend on gcd conditions for the chosen parameters.

5) Can this tool handle non-abelian groups?

This version focuses on finite additive (abelian) groups such as Z_n and their direct products. For non-abelian groups, generator images must satisfy different relations, and multiplication replaces addition.

6) Why might the homomorphism check be “sampled”?

For large n or n·r, checking all pairs can be slow. The tool may test a structured sample of additions to detect issues quickly, while still reporting formula-based sizes and well-definedness checks.