Subgroup & Coset Finder
Explore the structure of finite groups. Automatically find cyclic subgroups and visualize how cosets partition the group according to Lagrange’s Theorem.
Group Definition
Recommended: 2 to 40 for best visualization.
Visualization Area
Select parameters on the left and click a subgroup to visualize the partition.
Theory & Concepts
Lagrange’s Theorem
For any finite group $G$, the order (number of elements) of every subgroup $H$ divides the order of the group $G$.
$$ |G| = [G:H] \cdot |H| $$
Where $[G:H]$ is the Index of $H$ in $G$, representing the number of distinct cosets.
What is a Coset?
Given a subgroup $H$ and an element $g \in G$, the Left Coset $gH$ is the set:
$$ gH = \{ g * h \mid h \in H \} $$
The visualization above demonstrates that distinct cosets are disjoint (no overlap) and their union covers the entire group $G$.
Frequently Asked Questions
What are Cyclic Subgroups?
A cyclic subgroup is generated by a single element $a$. It consists of all powers (or multiples in additive groups) of $a$: $\langle a \rangle = \{ a, a^2, a^3, \dots, e \}$. In this tool, we automatically find all such subgroups for you.
Why is the Identity always in H?
By definition, a subgroup must contain the identity element. Consequently, one of the cosets is always the subgroup itself (usually written as $eH = H$).
Why do U(n) groups sometimes have fewer elements?
$U(n)$ contains only integers between 1 and $n$ that are coprime to $n$. For example, in $U(10)$, the elements are $\{1, 3, 7, 9\}$. The order is $\phi(10) = 4$, not 10.