Generators

Theorem: The intersection of subgroups $H_1, H_2, ....$ is a subgroup of each of $H_1, H_2, ...$

We say the elements $g_1,...,g_m$ are independent if none of them can be expressed in terms of the others, that is, $g_i \notin \langle g_1,...,g_{i-1},g_{i+1},...,g_m \rangle$. Clearly every finite group has at least one set of independent generators. Independent elements can have relations between them, e.g. if $a,b$ are independent then we may have $(a b)^2 = 1$ for example. Such a relation is called a defining relation.

Given any two groups $G, H$ we may form their direct product $G\times H$, whose elements are pairs $(g,h)$ with $g\in G, h\in H$, and the group operation applies coordinatewise. The direct product of abelian groups is abelian.

Suppose every element of a group $F$ has the form $g h$ where $g\in G, h\in H$ for some subgroups $G, H$ of $F$, and furthermore, suppose every element of $G$ commutes with every element of $H$ and $G \cap H = \{1\}$. Then $F \cong G\times H$.

It is clear how to generalize this to define the direct product to $k$ groups.

Example: $\mathbb{Z}_{15}^* \cong \mathbb{Z}_4 \times \mathbb{Z}_2$.