Normal Subgroup Checker Calculator

Inputs

Reset

Choose a mode. For table mode, element order must match table rows and columns.

Mode

Run associativity check

Verify associativity (may be slower)

For Zₙ, associativity is guaranteed.

Subgroup elements (optional override)

If provided, this list overrides generator input.

Zₙ Builder

n (group order)

Elements are 0 to n−1.

Step k (generator)

Subgroup is ⟨k⟩ under addition mod n.

Tip

In commutative groups, every subgroup is normal. This mode is ideal for quick coset and index exploration.

After submission, the computed result appears above this form, below the header.

Example Data Table

These examples show typical inputs and expected outcomes.

Scenario	Input	Subgroup	Expected Normal?	Note
Z₈ under addition	n=8, k=2	{0,2,4,6}	Yes	Commutative group, all subgroups are normal.
Z₁₂ under addition	n=12, k=4	{0,4,8}	Yes	Index is 4, cosets partition the group.
Non-commutative example	Table-based group	Any reflection-like subgroup	Sometimes No	Use the witness line to see the failing conjugation.

Formula Used

A subgroup H of a group G is normal when it is invariant under conjugation: gHg⁻¹ = H for every g ∈ G.

The checker tests the equivalent condition: for every g ∈ G and h ∈ H, the element g*h*g⁻¹ must lie in H. If not, it returns a counterexample witness.

How to Use This Calculator

Select a mode: Zₙ builder or Cayley table.
Enter group inputs and define the subgroup.
Enable associativity checking when needed.
Press Submit to compute normality and cosets.
Download CSV or PDF to save the findings.

FAQs

1) What does “normal” mean in this context?

A subgroup is normal when conjugating its elements by any group element keeps you inside the subgroup. This makes the quotient group well-defined.

2) Why are left and right cosets displayed?

Normality is closely tied to cosets. If the subgroup is normal, left cosets and right cosets coincide, giving cleaner structure and consistent partitions.

3) What is the counterexample witness?

If H is not normal, the tool shows specific g and h where g*h*g⁻¹ lands outside H. That single failure proves non-normality immediately.

4) Do I need to verify group axioms myself?

Table mode can detect identity and inverses, and it can test associativity when enabled. If those checks fail, the table may not define a valid group.

5) Why is Zₙ mode always normal for subgroups?

Zₙ under addition is commutative. In commutative groups, conjugation does nothing, so every subgroup is normal by definition.

6) What is Core(H) and why is it shown?

Core(H) is the intersection of all conjugates gHg⁻¹. It is the largest normal subgroup contained in H, useful when H is not normal.

7) How should I format the Cayley table?

Use one row per group element in your chosen order. Each row must have the same number of entries. Separate entries with spaces or commas.

8) What sizes are practical for table mode?

Smaller tables are best. This page limits table mode to 40 elements for reliability, while still allowing meaningful examples and fast checks.