External Direct Product Inverses/General Result
Theorem
Let $\ds \struct {\SS, \circ} = \prod_{k \mathop = 1}^n S_k$ be the external direct product of the algebraic structures $\struct {S_1, \circ_1}, \struct {S_2, \circ_2}, \ldots, \struct {S_n, \circ_n}$.
Let $\tuple {x_1, x_2, \ldots, x_n} \in S$.
Let $y_k$ be an inverse of $x_k$ in $\struct {S_k, \circ_k}$ for each of $k \in \N^*_n$.
Then $\tuple {y_1, y_2, \ldots, y_n}$ is the inverse of $\tuple {x_1, x_2, \ldots, x_n} \in \SS$ in $\struct {\SS, \circ}$.
Proof
Let $e_1, e_2, \ldots, e_n$ be the identity elements of $\struct {S_1, \circ_1}, \struct {S_2, \circ_2}, \ldots, \struct {S_n, \circ_n}$ respectively.
Let $x := \tuple {x_1, x_2, \ldots, x_n}$.
Let $y := \tuple {x_1, x_2, \ldots, x_n}$.
From External Direct Product Identity, $e := \tuple {e_1, e_2, \ldots, e_n}$ is the identity element of $\SS$.
Then:
\(\ds x \circ y\) | \(=\) | \(\ds \tuple {x_1, x_2, \ldots, x_n} \circ \tuple {y_1, y_2, \ldots, y_n}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \tuple {x_1 \circ_1 y_1, x_2 \circ_2 y_2, \ldots, x_n \circ_n y_n}\) | Definition of External Direct Product | |||||||||||
\(\ds \) | \(=\) | \(\ds \tuple {e_1, e_2, \ldots, e_n}\) | Definition of Inverse Element | |||||||||||
\(\ds \) | \(=\) | \(\ds e\) | External Direct Product Identity |
and:
\(\ds y \circ x\) | \(=\) | \(\ds \tuple {y_1, y_2, \ldots, y_n} \circ \tuple {x_1, x_2, \ldots, x_n}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \tuple {y_1 \circ_1 x_1, y_2 \circ_2 x_2, \ldots, y_n \circ_n x_n}\) | Definition of External Direct Product | |||||||||||
\(\ds \) | \(=\) | \(\ds \tuple {e_1, e_2, \ldots, e_n}\) | Definition of Inverse Element | |||||||||||
\(\ds \) | \(=\) | \(\ds e\) | External Direct Product Identity |
$\blacksquare$
Also see
- External Direct Product Associativity
- External Direct Product Commutativity
- External Direct Product Identity
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 18$: Induced $N$-ary Operations: Theorem $18.10: \ 3^\circ$