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Important case types for abelian groups
| <math>M</math> is uniquely <math>p</math>-divisible, i.e., every element of <math>M</math> can be divided by <matH>p</math> uniquely. This includes the case that <math>M</math> is a field of characteristic not 2. || all zero groups || all zero groups
| <math>M</math> is <math>p</math>-torsion-free, i.e., no nonzero element of <math>M</math> multiplies by <math>p</math> to give zero. || <math>(M/pM)^{(q-31)/2}</math> || <math>(M/pM)^{(q+2)/2}</math>
| <math>M</math> is <math>p</math>-divisible, but not necessarily uniquely so, e.g., <math>M = \mathbb{Q}/\mathbb{Z}</math> || <math>(\operatorname{Ann}_M(p))^{(q+3)/2}</math> || <math>(\operatorname{Ann}_M(p))^{q/2}</math>
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