External and Internal Direct Products Review

We will now review some of the recent material regarding external and internal direct products of groups.

• On The External Direct Product of Two Groups we defined the External Direct Product of the groups $(G, *)$ and $(H, +)$ to be the set $G \times H$ paired with the operation $\cdot$ defined for all $(g_1, h_1), (g_2, h_2) \in G \times H$ by:

(1)

\begin{align} \quad (g_1, h_1) \cdot (g_2, h_2) = (g_1 * g_2, h_1 + h_2) \end{align}

• We proved that $(G \times H, \cdot)$ is indeed a group.

• On The Order of an Element in an External Direct Product of Two Groups page we proved an important result. We proved that if $(G, *)$ and $(H, +)$ are groups and $(G \times H, \cdot)$ is the external direct product of these groups then the order of an element $(g, h) \in G \times H$ is:

(2)

\begin{align} \quad \mathrm{order} (g, h) = \mathrm{lcm} \left ( \mathrm{order} (g), \mathrm{order} (h) \right ) \end{align}

• On The Internal Direct Product of Two Groups page we defined another type of product on groups. We said that if $(G, *)$ is a group and $(H, *)$ and $(K, *)$ are subgroups, then $(G, *)$ is said to be the Internal Direct Product of $(H, *)$ and $(K, *)$ if:
  • (1) $G = \{ h * k : h \in H, \: k \in K \}$.
  • (2) $H \cap K = \{ e \}$ where $e$ is the identity element.
  • (3) $h * k = k * h$ for all $h \in H$ and for all $k \in K$.

• On the Internal Direct Products Isomorphic to External Direct Products we proved an important result regarding internal and external direct products. We proved that if $(G, *)$ is a group, $(H, *)$ and $(K, *)$ are subgroups, and $(G, *)$ is the internal direct product of $(H, *)$ and $(K, *)$ then:

(3)

\begin{align} \quad G \cong H \times K \end{align}

• We proved this by showing that the function $f : H \times K \to G$ defined for all $(h, k) \in H \times K$ by $f(h, k) = h * k$ is an isomorphism between $H \times K$ and $G$.