‘Atoms of symmetry’ introduces simple groups, which are the indivisible fragments of finite symmetry groups. A comparison can be made with prime numbers, but molecular structure provides a more apt analogy. The same simple groups (‘atoms’) give different finite groups (‘molecules’) if they are arranged in a different order. Examples of simple groups include cyclic groups, sporadic groups, and Mathieu groups. The complexity of the finite simple groups is testament to the power of mathematics, and applications include the link between simple groups and prime numbers in group theory, and the use of expander graphs in computer science.