Definition:Decomposable Group
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Definition
Let $\struct {G, \circ}$ be a group.
Then $\struct {G, \circ}$ is decomposable if and only if there exists a decomposition of $\struct {G, \circ}$.
That is, if and only if $\struct {G, \circ}$ is the internal direct product of two (or more) proper subgroups of $G$.
Indecomposable
$\struct {G, \circ}$ is indecomposable if and only if it is not decomposable.
That is, if and only if there does not exist a decomposition of $\struct {G, \circ}$.
Also see
- Results about decomposable groups can be found here.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings: Exercise $14.16$