Group Theory, Part 3: Direct and Semidirect Products

October 26th, 2017

Direct Products

Definition: For a finite number of groups $G_i$, $i \leq n$, the direct product $G=G_1 \times G_2 \times \dots \times G_n$ is the direct product of sets, with multiplication defined componentwise.

Evidently, each $G_i \trianglelefteq G$ in a direct product, as it is exactly the kernel of the projection which erases the $i$th coordinate. For a particular example, note that:

$$Z(G_1 \times G_2 \times \dots \times G_n) = Z(G_1)\times \dots \times Z(G_n)$$

In particular, note that a group is abelian iff $Z(G) =G$ (i.e. every element commutes with every other element). The above tells us that a direct product is abelian iff each of its factors is abelian.

The Fundamental Theorem of Finitely Generated Abelian Groups

Definition: A group $G$ is finitely generated if it is the span of a finite number of elements.

Definition: We call $\mathbb{Z}$ a free abelian group. A free abelian group of rank $r$ is the direct product $\mathbb{Z}\times\mathbb{Z} \times \dots \times \mathbb{Z}^r \cong \mathbb{Z}^r$.

The Fundamental Theorem of Finitely Generated Abelian Groups

Let $G$ be a finitely generated abelian group. Then:

$$G \cong \mathbb{Z}^r \times \mathbb{Z}_{n_1} \times \dots \times \mathbb{Z}_{n_s}$$

Such that $n_{i+1} \mid n_i$ for each $i$, $r \geq 0$ and each $n_j \geq 2$. Furthermore, the expression is unique.

This is really a consequence of the structure theorem for finitely generated modules over a PID, which is covered later on.

Definition: The number $r$ above is called the free rank or Betti number of $G$. The integers $n_i$ are called the invariant factors of $G$; the above is called the invariant factor decomposition of $G$.

Using the above theorem, we can indeed classify all the finitely generated abelian groups of a given order $n$. The problem is reduced to finding a sequence $n_1, \dots n_s$ of integers such that:

$$n_1 \dots n_s = n$$

And furthermore:

$$n_{i+1} \mid n_i$$

Note that each $n_i$ divides $n_i$; so in particular, $n_i$ contains every prime factor of $n$.

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