Mathematics · Discrete Mathematics

Homomorphism Kernel–Image Order Calculator

Calculate domain group order from kernel order and image order.

Runs locally
Your numbers

Inputs and results stay in this browser. Change one value at a time to explore the relationship.

Your inputCalculatedPassed forward in chains
domain group order120

Calculation steps

  1. Use c=ab with kernel order=6 and image order=20.
  2. domain group order=120.

Understand Homomorphism Kernel–Image Order

One idea, three depths

Choose how deeply to explain Homomorphism Kernel–Image Order

Homomorphism Kernel–Image Order: Calculate domain group order from kernel order and image order.

Age 5Explain it to a 5-year-oldStart with a picture

Imagine using Homomorphism Kernel–Image Order to answer this question: calculate domain group order from kernel order and image order? Enter kernel order and image order; the calculator shows domain group order. For example: kernel order=6 and image order=20 produce domain group order=120. The answer tells you domain group order.

Age 15Explain it to a 15-year-oldConnect it to the formula

For a finite group homomorphism, domain order equals kernel order times image order. This page evaluates the relationship directly. The rule is c=ab. Its input values are kernel order, image order, and the main result is domain group order. For example: kernel order=6 and image order=20 produce domain group order=120.

CollegeExplain it at college levelState the model precisely

This calculator evaluates the stated homomorphism kernel–image order relation over the valid real-number domain stated below. The implemented relation is c=ab, evaluated from kernel order, image order to produce domain group order. For a finite group homomorphism, domain order equals kernel order times image order. This page evaluates the relationship directly. The identity follows from the first isomorphism theorem.

Inputs and valid domain

  • kernel order must be a finite real number.
  • image order must be a finite real number.

Important boundary: The identity follows from the first isomorphism theorem.

The formula

c=ab

How the calculator works through it

It substitutes kernel order, image order into the formula and exposes every numerical step above. The main output is domain group order.

Read the result correctly

The domain group order is the direct answer to “calculate domain group order from kernel order and image order.” Read it with the units shown beside the inputs; a sign, angle, percentage or rate changes what the number means.

A worked check

kernel order=6 and image order=20 produce domain group order=120.

Where this model stops being reliable

The identity follows from the first isomorphism theorem.

Learn it by changing one value

Begin with the worked example, then change one value while keeping the others fixed. Compare the new result and calculation steps to identify which part of the formula changed.

Dictionary terms behind this calculator

Before studying the codeWhat you should know firstUse the calculator immediately, or check the foundations before reading the implementation.

These foundations help you understand why Homomorphism Kernel–Image Order works. They never block the calculator, and “optional” means useful context rather than a hidden requirement.

Hard requirements

  • Reading formulas and substituting values

    Homomorphism Kernel–Image Order uses c=ab. You need to recognise what each side represents before substituting the stated inputs or rearranging the relationship.

    Review this foundation about 4 min

Strong support

  • Sets, membership and finite collections

    Sets provide the objects and membership rules that give Homomorphism Kernel–Image Order its discrete meaning.

    Review this foundation about 6 min

Optional enrichment

Learn the missing foundationsI already know these — show the code

Mathematics → algorithm → program

Implement this calculation in code

These are direct reference implementations of the calculator's principal relationship and first output. They run locally and include a small known-answer check where the language supports it.

Algorithm

  1. Read kernel order, image order.
  2. Evaluate the principal relationship: c=ab.
  3. Return domain group order and check the domain conditions described above.
Python
            from math import *

def group_kernel_image_order_calculator(a, b) -> float:
    return (a * b)

assert abs(group_kernel_image_order_calculator(6, 20) - 120) < 1e-6 * max(1.0, abs(120))
          
Current calculator valuesUpdates when you change an input above.
              
            
C
            #include <assert.h>
#include <math.h>

double group_kernel_image_order_calculator(double a, double b) {
    return (a * b);
}

int main(void) {
    const double expected = 120;
    const double actual = group_kernel_image_order_calculator(6, 20);
    assert(fabs(actual - expected) < 1e-6 * fmax(1.0, fabs(expected)));
}
          
Current calculator valuesUpdates when you change an input above.
              
            
C++
            #include <cassert>
#include <cmath>
#include <numbers>

double group_kernel_image_order_calculator(double a, double b) {
    return (a * b);
}

int main() {
    constexpr double expected = 120;
    const double actual = group_kernel_image_order_calculator(6, 20);
    assert(std::fabs(actual - expected) < 1e-6 * std::fmax(1.0, std::fabs(expected)));
}
          
Current calculator valuesUpdates when you change an input above.
              
            
Linux x86-64 assembly

x86-64 NASM · System V ABI · Linux · SSE2 with libm where required

            ; double group_kernel_image_order_calculator(double a, double b)
; Linux x86-64 NASM · System V ABI · first eight doubles in xmm0–xmm7
global group_kernel_image_order_calculator
section .text

group_kernel_image_order_calculator:
    push rbp
    mov rbp, rsp
    sub rsp, 32
    movsd [rbp-8], xmm0
    movsd [rbp-16], xmm1
    movsd xmm0, [rbp-8]
    mulsd xmm0, [rbp-16]
    movsd [rbp-24], xmm0
    movsd xmm0, [rbp-24]
    leave
    ret
          
Current calculator valuesUpdates when you change an input above.
              
            
MATLAB
            function result = group_kernel_image_order_calculator(a, b)
    result = (a * b);
end
          
Current calculator valuesUpdates when you change an input above.
              
            
Wolfram Language
            ClearAll[mwCalculate];
mwCalculate[a_, b_] := (a * b);
          
Current calculator valuesUpdates when you change an input above.
              
            

Continue in mathematical software

The downloaded file includes your current inputs and first calculated result. It is created locally.

Floating-point answers can differ slightly by language, compiler and processor. Compare within a suitable tolerance rather than assuming every decimal representation will be identical.

Reuse the page responsiblyCite this pageAPA, MLA, Chicago, Harvard, BibTeX and RIS

These formats cite this calculator page itself. They are separate from the academic references above, which support the mathematical method and terminology.

APA 7

MW SysArc. (2026, July 21). Homomorphism Kernel–Image Order Calculator. MW SysArc Tools. https://math.mwsysarc.com/discrete-mathematics/group-kernel-image-order-calculator

MLA 9

MW SysArc. “Homomorphism Kernel–Image Order Calculator.” MW SysArc Tools, 21 July 2026, https://math.mwsysarc.com/discrete-mathematics/group-kernel-image-order-calculator. Accessed 31 Aug. 2026.

Chicago 17

MW SysArc. “Homomorphism Kernel–Image Order Calculator.” MW SysArc Tools. Published July 21, 2026. Accessed August 31, 2026. https://math.mwsysarc.com/discrete-mathematics/group-kernel-image-order-calculator.

Harvard

MW SysArc (2026) ‘Homomorphism Kernel–Image Order Calculator’, MW SysArc Tools. Published 21 July 2026. Available at: https://math.mwsysarc.com/discrete-mathematics/group-kernel-image-order-calculator (Accessed: 31 August 2026).

BibTeX and RIS records

BibTeX

@misc{mwsysarc_group_kernel_image_order_calculator_2026,
  author = {{MW SysArc}},
  title = {Homomorphism Kernel–Image Order Calculator},
  howpublished = {MW SysArc Tools},
  year = {2026},
  url = {https://math.mwsysarc.com/discrete-mathematics/group-kernel-image-order-calculator},
  note = {Published July 21, 2026; accessed August 31, 2026}
}

RIS

TY  - ELEC
AU  - MW SysArc
TI  - Homomorphism Kernel–Image Order Calculator
T2  - MW SysArc Tools
PY  - 2026
DA  - 2026-07-21
Y2  - 2026-08-31
UR  - https://math.mwsysarc.com/discrete-mathematics/group-kernel-image-order-calculator
N1  - Published July 21, 2026
ER  -

Clear answers

Frequently asked questions

What does the Homomorphism Kernel–Image Order do?

Calculate domain group order from kernel order and image order.

How does the Homomorphism Kernel–Image Order work?

The calculator applies c=ab. For a finite group homomorphism, domain order equals kernel order times image order. This page evaluates the relationship directly.

What can I learn from the Homomorphism Kernel–Image Order?

It connects the mathematical rule to your chosen numbers and shows each calculation step. Change one input at a time to see how the result responds.

Does MW SysArc receive or store what I enter?

No. The calculation runs locally in your browser. MW SysArc does not receive or store your calculation inputs.

How should I use the result?

Use the steps to understand the method, then verify important school or professional work using the notation and rounding rules required in your setting.

Last reviewed . Calculations tested .

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