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Oscar Plaisant
2024-09-20 21:31:23 +02:00
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démonstration le produit direct de groupes conserve la commutativité.md Normal file
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up:: [[produit direct de groupes abéliens]]
#maths/algèbre
$$
\begin{align}
G \times H \text{ est abélien} & \iff \forall (g, h), (g', h') \in G\times H, \quad (g, h)\times (g', h') = (g', h') \times (g, h)\\
&\iff \forall (g, h), (g', g') \in G\times H, \quad (g*_{G}g', h*_{H}h') = (g'*_{G}g, h'*_{H}h) \\
&\iff \forall (g, h), (g', g') \in G\times H, \quad g*_{G}g' = g'*_{G}g \wedge h*_{H}h' = h'*_{H}h \\
&\iff *_{G} \text{ et } *_{H} \text{ commutent} \\
&\iff G \text{ est abélien et } H \text{ est abélien} \\
&&\square
\end{align}
$$