Finding the Identity in different group structures
A word is a sequence of elements from a set of generators ($a$ and $b$). The Word Problem asks if a sequence simplifies to the Identity (the element that does "nothing").
In the Free Group, order is strict. In the Grid Group, $ab = ba$. In the Torus, $a^5 = e$ (Identity).
The Permutation Mode: This visualizes a group where $a$ and $b$ shuffle the positions of 4 colored nodes. Here, sequences can become "Identity" in unexpected ways. For example, applying a shuffle multiple times might eventually return all nodes to their original spots, meaning a non-trivial word like $aaaa$ might secretly be the Identity.