Tour:External direct product

This article adapts material from the main article: External direct product

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WHAT YOU NEED TO DO:

• Read, and understand, the definition of external direct product.
• Convince yourself that the external direct product of two groups satisfies the conditions for being a group (a quick written check of associativity, identity element and inverses may be good here).

Definition (for two groups)

Definition with symbols

Given two groups $G_{1}$ and $G_{2}$, the external direct product of $G_{1}$ and $G_{2}$, denoted as $G_{1}\times G_{2}$, is defined as follows:

• As a set, it is the Cartesian product of $G_{1}$ and $G_{2}$, that is, it is the set of ordered pairs $(g_{1},g_{2})$ with the first member $g_{1}$ from $G_{1}$ and the second member $g_{2}$ from $G_{2}$.
• The group operations are defined coordinate-wise, that is:

Operation name	Description of operation in terms of description of operations on factor groups	Explanation
Multiplication or product	$(g_{1},g_{2})(h_{1},h_{2})=(g_{1}h_{1},g_{2}h_{2})$ where $g_{1},h_{1}\in G_{1}$, $g_{2},h_{2}\in G_{2}$	We carry out the multiplication separately in each coordinate.
Identity element (or neutral element)	$\!e=(e_{1},e_{2})$ where $e_{1}$ is the identity element of $G_{1}$ and $e_{2}$ is the identity element of $G_{2}$.	To compute the identity element, we use the identity element in each coordinate.
Inverse map	$(g_{1},g_{2})^{-1}=(g_{1}^{-1},g_{2}^{-1})$	We carry out the inversion separately in each coordinate.

This page is part of the Groupprops guided tour for beginners (Jump to beginning of tour)
PREVIOUS: Cayley's theorem| UP: Introduction five (beginners)| NEXT: Product of subgroups
General instructions for the tour | Pedagogical notes for the tour | Pedagogical notes for this part