Every whole number bigger than 1 breaks apart into primes, and the primes themselves refuse to break any further — they are the atoms of multiplication. Group theory has its own atoms. A finite group can often be taken apart into smaller pieces using normal subgroups and quotient groups, but eventually you reach groups that cannot be broken down at all. Those un-breakable groups are called simple, and this page is about them and about the groups you can build by stacking simple pieces on top of one another.

This turns out to be one of the most consequential ideas in all of algebra. It answers a question that stumped mathematicians for three centuries: why is there a quadratic formula, a cubic formula, even a (monstrous) quartic formula — but no formula in radicals for the general quintic? The answer is not about the polynomials at all. It is about whether a certain group is solvable. Let's build up to that.

Simple groups: the atoms

Why does this deserve the word simple? Because a normal subgroup is exactly what lets you form a quotient G/N and so express G in terms of smaller groups. A simple group offers no such foothold: there is nothing to divide out by, so it cannot be reduced. It is a dead end for the "take a quotient" operation — an atom.

The smallest examples are hiding in plain sight. Take a cyclic group \mathbb{Z}_p of prime order p. By Lagrange's theorem every subgroup's order divides p, and the only divisors of a prime are 1 and p — so the only subgroups at all are \{e\} and the whole group. There is no room for anything in between, so \mathbb{Z}_p is simple.

The non-abelian atoms: the alternating groups

If the story stopped at \mathbb{Z}_p, simple groups would be a curiosity. The drama starts once we allow non-commuting atoms. The first and most famous family comes from the alternating group A_n — the even permutations of n objects.

We will take this on trust rather than prove it (the proof is a lovely but lengthy hunt through 3-cycles). The point to hold onto is that A_5 is an atom that does not commute. That single fact is the crack through which the quintic escapes every possible formula, as we'll see.

Composition series: factoring a group into atoms

A number is factored by writing it as a product of primes. A group is factored by building a composition series: a chain of subgroups

where each G_i is normal in the next one (G_i \triangleleft G_{i+1}) and every successive quotient G_{i+1}/G_i is simple. Those quotients are the composition factors — the atoms your group is made of. The number of steps k is the composition length.

Watch it happen for \mathbb{Z}_{12}. One valid chain climbs through the subgroups \{0\} \triangleleft \{0,6\} \triangleleft \{0,3,6,9\} \triangleleft \mathbb{Z}_{12}, and the three quotients along the way come out simple:

The composition factors are \mathbb{Z}_2, \mathbb{Z}_2, \mathbb{Z}_3 — and notice that 2 \times 2 \times 3 = 12 = |\mathbb{Z}_{12}|. Just like 12 = 2 \times 2 \times 3 as a number, the group's atoms multiply back to its order. The figure below builds this tower one floor at a time.

Jordan–Hölder: the atoms are unique

You might worry that a different chain of subgroups could give a different list of atoms — after all, \mathbb{Z}_{12} has other composition series. It cannot. This is the group theorist's version of the fundamental theorem of arithmetic.

So a group determines its atoms as firmly as 12 determines \{2, 2, 3\}. The order you discover them in may differ, and — a subtlety worth flagging — unlike numbers, the atoms alone do not always determine the group (different groups can share the same composition factors). But the factors themselves are a genuine invariant, a fingerprint every finite group carries.

Solvable groups: when all the atoms are abelian

Now for the star of the show. A group is solvable when it can be built out of commutative pieces.

Our tower for \mathbb{Z}_{12} had factors \mathbb{Z}_2, \mathbb{Z}_2, \mathbb{Z}_3 — all cyclic of prime order — so \mathbb{Z}_{12} is solvable. In fact the easy cases are very easy:

• Every abelian group is solvable (the one-step series \{e\} \triangleleft G already has an abelian quotient).
• Every p-group is solvable (a group whose order is a power of a prime).
• Products, subgroups and quotients of solvable groups are solvable.

The symmetric groups: solvable until they suddenly aren't

The symmetric group S_n of all permutations of n objects is where solvability lives or dies. For small n everything is fine:

• S_3 (order 6) has the series \{e\} \triangleleft A_3 \triangleleft S_3 with quotients A_3 \cong \mathbb{Z}_3 and S_3/A_3 \cong \mathbb{Z}_2 — both abelian. Solvable.
• S_4 (order 24) has the longer series \{e\} \triangleleft V_4 \triangleleft A_4 \triangleleft S_4 (where V_4 is the Klein four-group), with quotients of orders 4, 3, 2 — again all abelian. Solvable.

And then it breaks. For n \ge 5, S_n is not solvable. The reason is precisely the atom we met earlier: any subnormal series for S_5 runs into A_5, which is simple and non-abelian. Being simple, A_5 cannot be broken down further; being non-abelian, it cannot be one of the allowed abelian steps. It is an indivisible, non-commutative atom lodged inside S_5, and there is no way to route around it.

The punchline: why there is no quintic formula

Here is the bridge to Galois theory, and the reason this whole page matters. To every polynomial you can attach a group — its Galois group — that encodes the symmetries among its roots. The deep theorem is:

For a general quadratic, cubic or quartic, the Galois group is a subgroup of S_2, S_3 or S_4 — all solvable — so a formula in radicals exists (and people found them, painfully, over the Renaissance). But the general quintic has Galois group S_5, which we just saw is not solvable. By the theorem, no formula in radicals can possibly exist. Not "nobody has found one yet" — it is provably impossible, blocked by the single non-abelian atom A_5. A fact about a 60-element group settles a question about every quintic that will ever be written.