Advanced Right Coset Calculator

Calculator input

Use comma-separated labels. The Cayley table rows must follow the same element order shown in G.

Group elements G

List each element once in the exact table order.

Subgroup H

Enter a subset intended to be a subgroup.

Translation element g

The calculator forms the right coset H·g.

Cayley table

Each row represents left multiplication by its row element.

Also compute all distinct right cosets of H in G

Example data table

This example uses the additive group Z₆ with subgroup H = {0, 2, 4} and translation element g = 1.

Input set	Value	Computed output
Group G	{0, 1, 2, 3, 4, 5}	H·1 = {1, 3, 5}
Subgroup H	{0, 2, 4}
Translation element	1
Index	[G:H] = 2

Formula used

Right coset definition: H·g = { h·g | h ∈ H }

Coset size: |H·g| = |H| whenever H is a subgroup of G.

Index of H in G: [G:H] = |G| / |H|

Normality test: H is normal if g·H = H·g for every g in G.

For each subgroup element h, the calculator multiplies h by the chosen element g using the supplied Cayley table. Duplicate outputs are removed, then the result is ordered by the original group element list so the coset reads consistently.

How to use this calculator

Enter the finite group elements in one comma-separated line.
Paste the full Cayley table using the same order.
Enter a candidate subgroup H as a comma-separated subset.
Choose the element g that will translate H on the right.
Submit the form to validate the group and subgroup conditions.
Read the summary cards, product table, and optional distinct coset table.
Use the CSV and PDF buttons to export the computed report.

Frequently asked questions

1) What does a right coset mean?

A right coset shifts every element of a subgroup H by multiplying on the right with a fixed element g. The result is H·g = {h·g | h in H}.

2) Why do I need a Cayley table?

The table defines the group operation for every ordered pair of elements. Without it, the calculator cannot determine products like h·g or test subgroup closure accurately.

3) Why does the calculator validate the subgroup first?

Right cosets are defined relative to a subgroup. If H lacks the identity, closure, or inverses, the translated set may not have the standard structural properties expected from cosets.

4) Why can left and right cosets differ?

In non-abelian groups, multiplication order matters. That means g·H and H·g can contain different elements, so comparing both helps identify whether H is normal.

5) What is the index [G:H]?

The index counts how many distinct cosets of H partition G. For finite groups, it equals |G| divided by |H| when H is a valid subgroup.

6) Why are duplicate products removed?

A coset is a set, not a list. Sets do not repeat elements, so duplicates are removed before the final ordered coset is displayed.

7) Does every coset have the same size as H?

Yes. If H is a subgroup of G, every left or right coset of H has exactly |H| elements. The calculator checks subgroup conditions before reporting that structure.

8) What does the graph show?

The graph marks which listed group elements belong to H and which belong to the selected right coset H·g. It gives a quick visual comparison of translation.