Finite Groups: Cayley Tables and Subgroups
The whole structure of a finite group is in its Cayley table: the product of the row element and the column element (row $\cdot$ column) is in the cell where they meet. Every element appears exactly once in each row and each column. Here are all groups up to order $8$ (up to isomorphism) and $A_4$. Click a subgroup and only the products inside that subgroup stay highlighted in the table. Permutations are multiplied from left to right.
What to notice
- If the table is symmetric about the diagonal, the group is abelian. In $S_3$, $D_4$, $Q_8$ and $A_4$, find the cells where the symmetry breaks.
- Lagrange's theorem: the order of every subgroup divides the order of the group. The converse is false: $A_4$ has order $12$ but no subgroup with $6$ elements.
- The five groups with $8$ elements are told apart by their element orders: $Q_8$ has a single element of order $2$, $D_4$ has five.
- $Q_8$ is not abelian, yet all of its subgroups are normal.