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Basic subgroup
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<h2 id="definition-and-properties">Definition and properties</h2> <p>A <a href="/facts/Subgroup/r91mBgpa">subgroup</a>, <i>B</i>, of an <a href="/facts/Abelian_group/FfWwmx4R">abelian group</a>, <i>A</i>, is called <i>p</i>-basic, for a fixed <a href="/facts/Prime_number/L20FLkgn">prime number</a>, <i>p</i>, if the following conditions hold: </p> <ol><li><i>B</i> is a direct sum of <a href="/facts/Cyclic_group/JL5zfFuL">cyclic groups</a> of order <i>pn</i> and infinite cyclic groups;</li> <li><i>B</i> is a <i>p</i>-<a href="/facts/Pure_subgroup/MIvyNTcK">pure subgroup</a> of <i>A</i>;</li> <li>The quotient group, <i>A</i>/<i>B</i>, is a <i>p</i>-<a href="/facts/Divisible_group/1Q3QtFWe">divisible group</a>.</li></ol> <p>Conditions 1–3 imply that the subgroup, <i>B</i>, is <a href="/facts/Hausdorff_topological_space/gbKIrusl">Hausdorff</a> in the <i>p</i>-adic topology of <i>B</i>, which moreover coincides with the topology <a href="/facts/Induced_topology/xH9J6U04">induced</a> from <i>A</i>, and that <i>B</i> is <a href="/facts/Dense_subset/nPT9Yxao">dense</a> in <i>A</i>. Picking a generator in each cyclic direct summand of <i>B</i> creates a <i> </i>p<i>-basis</i> of <i>B</i>, which is analogous to a <a href="/facts/Basis_(linear_algebra)/89IPoN6c">basis</a> of a <a href="/facts/Vector_space/M1pxMLx2">vector space</a> or a <a href="/facts/Free_abelian_group/oaSRGmpY">free abelian group</a>. </p><p>Every abelian group, <i>A</i>, contains <i>p</i>-basic subgroups for each <i>p</i>, and any 2 <i>p</i>-basic subgroups of <i>A</i> are isomorphic. Abelian groups that contain a unique <i>p</i>-basic subgroup have been completely characterized. For the case of <a href="/facts/P-group/MHeGQT9S"><i>p</i>-groups</a> they are either <a href="/facts/Divisible_group/1Q3QtFWe">divisible</a> or <i>bounded</i>; i.e., have bounded exponent. In general, the isomorphism class of the quotient, <i>A</i>/<i>B</i> by a basic subgroup, <i>B</i>, may depend on <i>B</i>. </p> <ul><li>László Fuchs (1970), <i>Infinite abelian groups, Vol. I</i>. Pure and Applied Mathematics, Vol. 36. New York–London: Academic Press <a href="/facts/MR_(identifier)/uP137L11">MR</a><a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0255673">0255673</a></li> <li>L. Ya. Kulikov, <i>On the theory of abelian groups of arbitrary cardinality</i> (in Russian), Mat. Sb., 16 (1945), 129–162</li> <li><a href="/facts/Aleksandr_Gennadievich_Kurosh/ACYN3SPS">Kurosh, A. G.</a> (1960), <i>The theory of groups</i>, New York: Chelsea, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0109842">0109842</a></li></ul>