[[Group theory MOC]]
# Simple group

A **simple group** $G$ is a nontrivial [[group]] whose only [[Normal subgroup|normal subgroups]] are the [[trivial group]] and the group itself. #m/def/group
Such a group cannot be decomposed into the product of smaller groups.

## Examples

- $S_{3}$ is not simple, since it has a non-trivial normal subgroup $\{ e, (123), (132) \} \triangleleft S_{3}$.

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