Definition:Group Direct Product

From ProofWiki
Jump to: navigation, search

Definition

Let $\struct {G, \circ_1}$ and $\struct {H, \circ_2}$ be groups.

Let $G \times H: \set {\tuple {g, h}: g \in G, h \in H}$ be their cartesian product.


The (external) direct product of $\struct {G, \circ_1}$ and $\struct {H, \circ_2}$ is the group $\struct {G \times H, \circ}$ where the operation $\circ$ is defined as:

$\tuple {g_1, h_1} \circ \tuple {g_2, h_2} = \tuple {g_1 \circ_1 g_2, h_1 \circ_2 h_2}$


This is usually referred to as the group direct product of $G$ and $H$.


Finite Product

Let $\struct {G_1, \circ_1}, \struct {G_2, \circ_2}, \ldots, \struct {G_n, \circ_n}$ be groups.

Let $\displaystyle G = \prod_{k \mathop = 1}^n G_k$ be their cartesian product.


Let $\circ$ be the operation defined on $G$ as:

$\circ := \tuple {g_1, g_2, \ldots, g_n} \circ \tuple {h_1, h_2, \ldots, h_n} = \tuple {g_1 \circ_1 h_1, g_2 \circ_2 h_2, \ldots, g_n \circ_n h_n}$

for all ordered $n$-tuples in $G$.


The group $\struct {G, \circ}$ is called the (external) direct product of $\struct {G_1, \circ_1}, \struct {G_2, \circ_2}, \ldots, \struct {G_n, \circ_n}$.


General Definition

Let $\family {\struct {G_i, \circ_i} }_{i \mathop \in I}$ be a family of groups.

Let $\displaystyle G = \prod_{i \mathop \in I} G_i$ be their cartesian product.


Let $\circ$ be the operation defined on $G$ as:

$\circ := \family {g_i}_{i \mathop \in I} \circ \family {h_i}_{i \mathop \in I} = \family {g_i \circ_i h_i}_{i \mathop \in I}$

for all sequences in $G$.


The group $\struct {G, \circ}$ is called the (external) direct product of $\family {\struct {G_i, \circ_i} }_{i \mathop \in I}$.


Examples

Cyclic Group $C_2$ by Itself

The direct product of $C_2$, the cyclic group of order $2$, with itself is as follows.


Let us represent $C_2$ as the group $\struct {\set {1, -1}, \times}$:

$\begin {array} {r|rr} \struct {\set {1, -1} , \times} & 1 & -1 \\ \hline 1 & 1 & -1 \\ -1 & -1 & 1 \\ \end{array}$


Then the Cayley table for $C_2 \times C_2$ can be portrayed as:

$\begin {array} {c|cccc} C_2 \times C_2 & \tuple { 1, 1} & \tuple { 1, -1} & \tuple {-1, 1} & \tuple {-1, -1} \\ \hline \tuple { 1, 1} & \tuple { 1, 1} & \tuple { 1, -1} & \tuple {-1, 1} & \tuple {-1, -1} \\ \tuple { 1, -1} & \tuple { 1, -1} & \tuple { 1, 1} & \tuple {-1, -1} & \tuple {-1, 1} \\ \tuple {-1, 1} & \tuple {-1, 1} & \tuple {-1, -1} & \tuple { 1, 1} & \tuple { 1, -1} \\ \tuple {-1, -1} & \tuple {-1, -1} & \tuple {-1, 1} & \tuple { 1, -1} & \tuple { 1, 1} \\ \end{array}$


Cyclic Group $C_2$ by $C_3$

The direct product of $C_2$, the cyclic group of order $2$, with $C_3$, the cyclic group of order $3$, is as follows.


Let us represent $C_2$ as the group $\struct {\Z_2, +_2}$:

$\begin {array} {r|rr} +_2 & \eqclass 0 2 & \eqclass 1 2 \\ \hline \eqclass 0 2 & \eqclass 0 2 & \eqclass 1 2 \\ \eqclass 1 2 & \eqclass 1 2 & \eqclass 0 2 \\ \end{array}$


and $C_3$ as the group $\struct {\Z_3, +_3}$:

$\begin {array} {r|rrr} +_3 & \eqclass 0 3 & \eqclass 1 3 & \eqclass 2 3 \\ \hline \eqclass 0 3 & \eqclass 0 3 & \eqclass 1 3 & \eqclass 2 3 \\ \eqclass 1 3 & \eqclass 1 3 & \eqclass 2 3 & \eqclass 0 3 \\ \eqclass 2 3 & \eqclass 2 3 & \eqclass 0 3 & \eqclass 1 3 \\ \end{array}$


Then the Cayley table for $\struct{C_2 \times C_3, +_6}$ can be portrayed as:

$\begin {array} {r|rrrrrr} +_6 & \tuple {\eqclass 0 2, \eqclass 0 3} & \tuple {\eqclass 0 2, \eqclass 1 3} & \tuple {\eqclass 0 2, \eqclass 2 3} & \tuple {\eqclass 1 2, \eqclass 0 3} & \tuple {\eqclass 1 2, \eqclass 1 3} & \tuple {\eqclass 1 2, \eqclass 2 3} \\ \hline \tuple {\eqclass 0 2, \eqclass 0 3} & \tuple {\eqclass 0 2, \eqclass 0 3} & \tuple {\eqclass 0 2, \eqclass 1 3} & \tuple {\eqclass 0 2, \eqclass 2 3} & \tuple {\eqclass 1 2, \eqclass 0 3} & \tuple {\eqclass 1 2, \eqclass 1 3} & \tuple {\eqclass 1 2, \eqclass 2 3} \\ \tuple {\eqclass 0 2, \eqclass 1 3} & \tuple {\eqclass 0 2, \eqclass 1 3} & \tuple {\eqclass 0 2, \eqclass 2 3} & \tuple {\eqclass 0 2, \eqclass 0 3} & \tuple {\eqclass 1 2, \eqclass 1 3} & \tuple {\eqclass 1 2, \eqclass 2 3} & \tuple {\eqclass 1 2, \eqclass 0 3} \\ \tuple {\eqclass 0 2, \eqclass 2 3} & \tuple {\eqclass 0 2, \eqclass 2 3} & \tuple {\eqclass 0 2, \eqclass 0 3} & \tuple {\eqclass 0 2, \eqclass 1 3} & \tuple {\eqclass 1 2, \eqclass 2 3} & \tuple {\eqclass 1 2, \eqclass 0 3} & \tuple {\eqclass 1 2, \eqclass 1 3} \\ \tuple {\eqclass 1 2, \eqclass 0 3} & \tuple {\eqclass 1 2, \eqclass 0 3} & \tuple {\eqclass 1 2, \eqclass 1 3} & \tuple {\eqclass 1 2, \eqclass 2 3} & \tuple {\eqclass 0 2, \eqclass 0 3} & \tuple {\eqclass 0 2, \eqclass 1 3} & \tuple {\eqclass 0 2, \eqclass 2 3} \\ \tuple {\eqclass 1 2, \eqclass 1 3} & \tuple {\eqclass 1 2, \eqclass 1 3} & \tuple {\eqclass 1 2, \eqclass 2 3} & \tuple {\eqclass 1 2, \eqclass 0 3} & \tuple {\eqclass 0 2, \eqclass 1 3} & \tuple {\eqclass 0 2, \eqclass 2 3} & \tuple {\eqclass 0 2, \eqclass 0 3} \\ \tuple {\eqclass 1 2, \eqclass 2 3} & \tuple {\eqclass 1 2, \eqclass 2 3} & \tuple {\eqclass 1 2, \eqclass 0 3} & \tuple {\eqclass 1 2, \eqclass 1 3} & \tuple {\eqclass 0 2, \eqclass 2 3} & \tuple {\eqclass 0 2, \eqclass 0 3} & \tuple {\eqclass 0 2, \eqclass 1 3} \\ \end{array}$


Cyclic Group $C_3$ by Itself

The direct product of $C_3$, the cyclic group of order $3$, with itself is as follows.


Let us represent $C_3$ as the group $\struct {\Z_3, +_3}$:

$\begin {array} {r|rrr} +_3 & \eqclass 0 3 & \eqclass 1 3 & \eqclass 2 3 \\ \hline \eqclass 0 3 & \eqclass 0 3 & \eqclass 1 3 & \eqclass 2 3 \\ \eqclass 1 3 & \eqclass 1 3 & \eqclass 2 3 & \eqclass 0 3 \\ \eqclass 2 3 & \eqclass 2 3 & \eqclass 0 3 & \eqclass 1 3 \\ \end{array}$


Then the Cayley table for $\struct{C_2 \times C_3, +_9}$ can be portrayed as:

$\begin {array} {r|rrrrrrrrr} +_{3, 3} & \tuple {\eqclass 0 3, \eqclass 0 3} & \tuple {\eqclass 0 3, \eqclass 1 3} & \tuple {\eqclass 0 3, \eqclass 2 3} & \tuple {\eqclass 1 3, \eqclass 0 3} & \tuple {\eqclass 1 3, \eqclass 1 3} & \tuple {\eqclass 1 3, \eqclass 2 3} & \tuple {\eqclass 2 3, \eqclass 0 3} & \tuple {\eqclass 2 3, \eqclass 1 3} & \tuple {\eqclass 2 3, \eqclass 2 3} \\ \hline \tuple {\eqclass 0 3, \eqclass 0 3} & \tuple {\eqclass 0 3, \eqclass 0 3} & \tuple {\eqclass 0 3, \eqclass 1 3} & \tuple {\eqclass 0 3, \eqclass 2 3} & \tuple {\eqclass 1 3, \eqclass 0 3} & \tuple {\eqclass 1 3, \eqclass 1 3} & \tuple {\eqclass 1 3, \eqclass 2 3} & \tuple {\eqclass 2 3, \eqclass 0 3} & \tuple {\eqclass 2 3, \eqclass 1 3} & \tuple {\eqclass 2 3, \eqclass 2 3} \\ \tuple {\eqclass 0 3, \eqclass 1 3} & \tuple {\eqclass 0 3, \eqclass 1 3} & \tuple {\eqclass 0 3, \eqclass 2 3} & \tuple {\eqclass 0 3, \eqclass 0 3} & \tuple {\eqclass 1 3, \eqclass 1 3} & \tuple {\eqclass 1 3, \eqclass 2 3} & \tuple {\eqclass 1 3, \eqclass 0 3} & \tuple {\eqclass 2 3, \eqclass 1 3} & \tuple {\eqclass 2 3, \eqclass 2 3} & \tuple {\eqclass 2 3, \eqclass 0 3} \\ \tuple {\eqclass 0 3, \eqclass 2 3} & \tuple {\eqclass 0 3, \eqclass 2 3} & \tuple {\eqclass 0 3, \eqclass 0 3} & \tuple {\eqclass 0 3, \eqclass 1 3} & \tuple {\eqclass 1 3, \eqclass 2 3} & \tuple {\eqclass 1 3, \eqclass 0 3} & \tuple {\eqclass 1 3, \eqclass 1 3} & \tuple {\eqclass 2 3, \eqclass 2 3} & \tuple {\eqclass 2 3, \eqclass 0 3} & \tuple {\eqclass 2 3, \eqclass 1 3} \\ \tuple {\eqclass 1 3, \eqclass 0 3} & \tuple {\eqclass 1 3, \eqclass 3 3} & \tuple {\eqclass 1 3, \eqclass 1 3} & \tuple {\eqclass 1 3, \eqclass 2 3} & \tuple {\eqclass 2 3, \eqclass 0 3} & \tuple {\eqclass 2 3, \eqclass 1 3} & \tuple {\eqclass 2 3, \eqclass 2 3} & \tuple {\eqclass 0 3, \eqclass 0 3} & \tuple {\eqclass 0 3, \eqclass 1 3} & \tuple {\eqclass 0 3, \eqclass 2 3} \\ \tuple {\eqclass 1 3, \eqclass 1 3} & \tuple {\eqclass 1 3, \eqclass 1 3} & \tuple {\eqclass 1 3, \eqclass 2 3} & \tuple {\eqclass 1 3, \eqclass 0 3} & \tuple {\eqclass 2 3, \eqclass 1 3} & \tuple {\eqclass 2 3, \eqclass 2 3} & \tuple {\eqclass 2 3, \eqclass 0 3} & \tuple {\eqclass 0 3, \eqclass 1 3} & \tuple {\eqclass 0 3, \eqclass 2 3} & \tuple {\eqclass 0 3, \eqclass 0 3} \\ \tuple {\eqclass 1 3, \eqclass 2 3} & \tuple {\eqclass 1 3, \eqclass 2 3} & \tuple {\eqclass 1 3, \eqclass 0 3} & \tuple {\eqclass 1 3, \eqclass 1 3} & \tuple {\eqclass 2 3, \eqclass 2 3} & \tuple {\eqclass 2 3, \eqclass 0 3} & \tuple {\eqclass 2 3, \eqclass 1 3} & \tuple {\eqclass 0 3, \eqclass 2 3} & \tuple {\eqclass 0 3, \eqclass 0 3} & \tuple {\eqclass 0 3, \eqclass 1 3} \\ \tuple {\eqclass 2 3, \eqclass 0 3} & \tuple {\eqclass 2 3, \eqclass 0 3} & \tuple {\eqclass 2 3, \eqclass 1 3} & \tuple {\eqclass 2 3, \eqclass 2 3} & \tuple {\eqclass 0 3, \eqclass 0 3} & \tuple {\eqclass 0 3, \eqclass 1 3} & \tuple {\eqclass 0 3, \eqclass 2 3} & \tuple {\eqclass 1 3, \eqclass 0 3} & \tuple {\eqclass 1 3, \eqclass 1 3} & \tuple {\eqclass 1 3, \eqclass 2 3} \\ \tuple {\eqclass 2 3, \eqclass 1 3} & \tuple {\eqclass 2 3, \eqclass 1 3} & \tuple {\eqclass 2 3, \eqclass 2 3} & \tuple {\eqclass 2 3, \eqclass 0 3} & \tuple {\eqclass 0 3, \eqclass 1 3} & \tuple {\eqclass 0 3, \eqclass 2 3} & \tuple {\eqclass 0 3, \eqclass 0 3} & \tuple {\eqclass 1 3, \eqclass 1 3} & \tuple {\eqclass 1 3, \eqclass 2 3} & \tuple {\eqclass 1 3, \eqclass 0 3} \\ \tuple {\eqclass 2 3, \eqclass 2 3} & \tuple {\eqclass 2 3, \eqclass 2 3} & \tuple {\eqclass 2 3, \eqclass 0 3} & \tuple {\eqclass 2 3, \eqclass 1 3} & \tuple {\eqclass 0 3, \eqclass 2 3} & \tuple {\eqclass 0 3, \eqclass 0 3} & \tuple {\eqclass 0 3, \eqclass 1 3} & \tuple {\eqclass 1 3, \eqclass 2 3} & \tuple {\eqclass 1 3, \eqclass 0 3} & \tuple {\eqclass 1 3, \eqclass 1 3} \\ \end{array}$


Also known as

The group direct product is referred to in some sources, when dealing with additive groups, as the (group) direct sum.

In such contexts, the symbol $G \times H$ can often be seen as $G \mathop {\dot +} H$.

On $\mathsf{Pr} \infty \mathsf{fWiki}$ we consider all groups, whatever their nature, to be instances of the same abstract concept, and therefore do not use this notation.


Warning

Note that $G$ and $H$, and so on, are not subsets of $G \times H$ and therefore are not subgroups of it either.

There exist subgroups in $G \times H$ which are isomorphic with $G$ and $H$ though, namely:

$G \times \set {e_H}$ and $\set {e_G} \times H$

where $e_G$ and $e_H$ are identity elements of $G$ and $H$ respectively.


Also see

  • Results about group direct products can be found here.


Generalizations


Sources