Simple groups are the building blocks of groups similar to the way prime numbers are the building blocks of integers. This post will unpack this analogy in two ways:
- How do simple groups compare to prime numbers?
- How does the composition of simple groups compare to the composition of prime numbers?
The former analogy is stronger than the latter.
Primes and simple groups
A simple group has no nontrivial subgroups, just a prime number has no nontrivial factors. Except that’s not quite right. A simple group is defined as having no nontrivial normal subgroups. The previous post compares normal and non-normal subgroups. Normal subgroups have nice properties which are necessary for decomposition and composition. You can’t define quotients for non-normal groups.
Every subgroup of an Abelian group is normal, so in the context of Abelian groups it is true that simple groups have no nontrivial subgroups, i.e. the only subgroups of a simple Abelian group G are the identity and G itself. It follows from Sylow’s theorems that the order of a finite Abelian group with no nontrivial factors must be an integer with no nontrivial factors, i.e. a prime number. Every Abelian finite simple group must be isomoprphic to the integers mod p for some prime p.
Non-Abelian finite simple groups do not have prime order, but they not decomposable in the sense described below.
Composition and decomposition
Prime numbers compose to form other numbers by products. You can also compose groups by taking products, though you need more than that. It is not the case that all finite groups are products of finite simple groups.
Let ℤn denote the cyclic group of order n and let ⊕ denote direct sum. The group ℤ4 is not isomorphic to ℤ2 ⊕ ℤ2. Even in the case of Abelian groups, not all Abelian groups are the direct sum or direct product of simple groups. [1]
Finite groups can be decomposed into smaller finite simple groups, but we can’t easily or uniquely rebuild a group from this decomposition.
The Jordan-Hölder theorem says that a finite group G has a composition series
1 = H0 ⊲ H1 ⊲ … ⊲ Hn = G
where each H is a maximal normal subgroup of the next, the quotients Hi+1 / Hi of consecutive are simple groups. The composition series is not unique, but all such series are equivalent in a sense that the Jordan-Hölder theorem makes precise.
It seems to me that the composition series ought to be called a decomposition series in that you can start with G and find the H‘s, but it’s a difficult problem, known as “the extension problem,” to reconstruct G from the H‘s, and in general there are multiple solutions.
The analogy to prime numbers would be if there was an essentially unique way to factor a number, but not a unique way to multiply the factors back together.
Reductionism
Some people thought that the classification of finite simple groups would be the end group theory. That has not been the case. Some also thought sequencing of the human genome would lead to cures for a huge range of diseases. That has not been the case either. Reductionism often produces disappointing results.
Related posts
[1] In the context of Abelian groups, (direct) products and coproducts (i.e. direct sums) are isomorphic.