External Direct Product Inverses/General Result

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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


Sources