18.1 An example: the Euclidean group

Concept links · terms present in this machine draft; source roles are unverified: group action

Given two groups $G^{'}$ and $G^{''}$ , a product group is formed by taking pairs of elements $(g^{'}, g^{''}) \in G^{'} \times G^{''}$ . However, when the two groups act on the same space, but elements of $G^{'}$ and $G^{''}$ don’t commute, a diferent sort of product group is needed to describe the group action. As an example, consider the case of pairs $(a_{2}, R_{2})$ of elements $a_{2} \in R^{3}$ and $R_{2} \in S O (3)$ , acting on $R^{3}$ by translation and rotation

v \to (a_{2}, R_{2}) \cdot v = a_{2} + R_{2} v

If one then acts on the result with $(a_{1}, R_{1})$ one gets

(a_{1}, R_{1}) \cdot ((a_{2}, R_{2}) \cdot v) = (a_{1}, R_{1}) \cdot (a_{2} + R_{2} v) = a_{1} + R_{1} a_{2} + R_{1} R_{2} v

Note that this is not what one would get if one took the product group law on $R^{3} \times S O (3)$ , since then the action of $(a_{1}, R_{1}) (a_{2}, R_{2})$ on $R^{3}$ would be

v \to a_{1} + a_{2} + R_{1} R_{2} v

To get the correct group action on $R^{3}$ , one needs to take $R^{3} \times S O (3)$ not with the product group law, but instead with the group law

(a_{1}, R_{1}) (a_{2}, R_{2}) = (a_{1} + R_{1} a_{2}, R_{1} R_{2})

This group law difers from the standard product law by a term $R_{1} a_{2}$ , which is the result of $R_{1} \in S O (3)$ acting non-trivially on $a_{2} \in R^{3}$ . We will denote the set $R^{3} \times S O (3)$ with this group law by

R^{3} ⋊ S O (3)

This is the group of orientation-preserving transformations of $R^{3}$ preserving the standard inner product.

The same construction works in arbitrary dimensions, where one has:

Definition (Euclidean group)

The Euclidean group $E (d)$ (sometimes written $I S O (d)$ for “inhomogeneous” rotation group) in dimension $d$ is the product of the translation and rotation groups of $R^{d}$ as a set, with multiplication law
$(a_{1}, R_{1}) (a_{2}, R_{2}) = (a_{1} + R_{1} a_{2}, R_{1} R_{2})$

(where $a_{j} \in R^{d}, R_{j} \in S O (d))$ and can be denoted $b y$

R^{d} ⋊ S O (d)

$E (d)$ can also be written as a matrix group, taking it to be the subgroup of $G L (d + 1, R)$ of matrices of the form ( $R$ is a d by $d$ orthogonal matrix, a a d dimensional column vector)

(R 0 a 1)

One gets the multiplication law for $E (d)$ from matrix multiplication since

(R_{1} 0 a_{1} 1) (R_{2} 0 a_{2} 1) = (R_{1} R_{2} 0 a_{1} + R_{1} a_{2} 1)

来源与版本

正文：英文 · OCR 机器稿 · 待校对

核对状态：OCR 机器稿 · 待校对

原书 PDF · 印刷页 204、205、206、207、208、209

来源版本：2025-10-20

来源 PDF SHA-256：5a1941b2443b54d5db3d055f1e5ba390429b7a728475258017aaac87ee85a837

OCR 来源 SHA-256：56b50a7927e16183e674d182a47fa8abf3c26f92b527480ec6a89fb14eeee0d6

OCR 产物 SHA-256：56b50a7927e16183e674d182a47fa8abf3c26f92b527480ec6a89fb14eeee0d6

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18.1 An example: the Euclidean group

18.1 An example: the Euclidean group

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