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Height (abelian group)
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the height of an element <i>g</i> of an <a href="/facts/Abelian_group/FfWwmx4R">abelian group</a> <i>A</i> is an invariant that captures its divisibility properties: it is the largest <a href="/facts/Natural_number/u65og8JE">natural number</a> <i>N</i> such that the equation <i>Nx</i> = <i>g</i> has a solution <i>x</i> ∈ <i>A</i>, or the symbol ∞ if there is no such <i>N</i>. The <i>p</i>-height considers only divisibility properties by the powers of a fixed <a href="/facts/Prime_number/L20FLkgn">prime number</a> <i>p</i>. The notion of height admits a refinement so that the <i>p</i>-height becomes an <a href="/facts/Ordinal_number/GelFLjJt">ordinal number</a>. Height plays an important role in <a href="/facts/Pr%25C3%25BCfer_theorems/f8R1Cpmb">Prüfer theorems</a> and also in Ulm's theorem, which describes the classification of certain infinite abelian groups in terms of their Ulm factors or Ulm invariants. </p>