Divisible Group

Examples

• Q is divisible, as additive abelian group
• More generally, every vector space over Q has a divisible underlying group.
• Every quotient of a divisible group is divisible. Thus, Q/Z is divisible.
• The p-primary component of Q/Z which is isomorphic to the p-quasicyclic group $Zp^\backslash infty$ is divisible.
• Every existentially closed group (in the model theoretic sense) is divisible.

Structure theorem of divisible groups

$G\; =\; Tor(G)\; \backslash oplus\; G/Tor(G)$.

$G\; =\; \backslash oplus\_\{i\; \backslash in\; I\}\; Q\; =\; Q^\{(I)\}$.

$(Tor(G))\_p\; =\; \backslash oplus\_\{i\; \backslash in\; I\_p\}\; Zp^\backslash infty=\; Zp^\backslash infty^\{(I\_p)\},$

$G\; =\; (\backslash oplus\_\{p\; \backslash in\; P\}\; Zp^\backslash infty^\{(I\_p)\})\; \backslash oplus\; Q^\{(I)\}$.

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