Splitting in Short exact sequence

abstract-algebra group-extensions

Is not split:

If it is we get $\Bbb R=\Bbb R/\Bbb Z\oplus \Bbb Z$ which is not possible, because there is no element of finite order in $\Bbb R$.


The sequence $0 \to \mathbb{Q}\to \mathbb{R} \to \mathbb{R}/\mathbb{Q}\to 0$ splits.

To have a split one you need the subgroup to be divisible ( a $\mathbb{Q}$ vector subspace).

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