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ZJ theorem
open-in-new
<h2 id="notation-and-definitions">Notation and definitions</h2> <ul><li><i>J</i>(<i>S</i>) is the <a href="/facts/Thompson_subgroup/IawxmTQJ">Thompson subgroup</a> of a <i>p</i>-group <i>S</i>: the subgroup generated by the <a href="/facts/Abelian_group/FfWwmx4R">abelian subgroups</a> of maximal <a href="/facts/Order_(group_theory)/Cl3tKGFg">order</a>.</li> <li><i>Z</i>(<i>H</i>) means the <a href="/facts/Center_(group_theory)/gqAg6B8I">center</a> of a group <i>H</i>.</li> <li><i>O</i><i>p′</i> is the maximal normal subgroup of <i>G</i> of order <a href="/facts/Coprime/bnCxCXxs">coprime</a> to <i>p</i>, the <a href="/facts/Core_(group_theory)/IGEJE92q"><i>p′</i>-core</a></li> <li><i>O</i><i>p</i> is the maximal normal <i>p</i>-subgroup of <i>G</i>, the <a href="/facts/P-core/IGEJE92q"><i>p</i>-core</a>.</li> <li><i>O</i><i>p′</i>,<i>p</i>(<i>G</i>) is the maximal normal <a href="/facts/P-nilpotent_group/t4mEQ3YS"><i>p</i>-nilpotent subgroup</a> of <i>G</i>, the <a href="/facts/Core_(group_theory)/IGEJE92q"><i>p′</i>,<i>p</i>-core</a>, part of the <a href="/facts/Upper_p-series/IGEJE92q">upper <i>p</i>-series</a>.</li> <li>For an odd prime <i>p</i>, a group <i>G</i> with <i>O</i><i>p</i>(<i>G</i>) ≠ 1 is said to be <a href="/facts/P-stable_group/MXVJNzUz"><i>p</i>-stable</a> if whenever <i>P</i> is a <i>p</i>-subgroup of <i>G</i> such that <i>POp′</i>(<i>G</i>) is normal in <i>G</i>, and [<i>P</i>,<i>x</i>,<i>x</i>] = 1, then the image of <i>x</i> in N<i>G</i>(<i>P</i>)/C<i>G</i>(<i>P</i>) is contained in a normal <i>p</i>-subgroup of N<i>G</i>(<i>P</i>)/C<i>G</i>(<i>P</i>).</li> <li>For an odd prime <i>p</i>, a group <i>G</i> with <i>O</i><i>p</i>(<i>G</i>) ≠ 1 is said to be <a href="/facts/P-constrained_group/CkHwtRNx"><i>p</i>-constrained</a> if the <a href="/facts/Centralizer/AKhrEUGR">centralizer</a> C<i>G</i>(<i>P</i>) is contained in <i>O</i><i>p′</i>,<i>p</i>(<i>G</i>) whenever <i>P</i> is a Sylow <i>p</i>-subgroup of <i>O</i><i>p′</i>,<i>p</i>(<i>G</i>).</li></ul> <ul><li><a href="/facts/George_Glauberman/PotfDt5q">Glauberman, George</a> (1968), <a href="http://www.cms.math.ca/cjm/v20/p1101">"A characteristic subgroup of a p-stable group"</a>, <i><a href="/facts/Canadian_Journal_of_Mathematics/oHhVvs1V">Canadian Journal of Mathematics</a></i>, 20: 1101–1135, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.4153%2Fcjm-1968-107-2">10.4153/cjm-1968-107-2</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0008-414X">0008-414X</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0230807">0230807</a></li> <li><a href="/facts/Daniel_Gorenstein/tBgFlDpt">Gorenstein, D.</a> (1980), <i>Finite Groups</i>, New York: Chelsea, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8284-0301-6, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0569209">0569209</a></li> <li><a href="/facts/John_G._Thompson/M6tF5WNE">Thompson, John G.</a> (1969), "A replacement theorem for p-groups and a conjecture", <i><a href="/facts/Journal_of_Algebra/Xr6PbIKM">Journal of Algebra</a></i>, 13 (2): 149–151, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2F0021-8693%2869%2990068-4">10.1016/0021-8693(69)90068-4</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0021-8693">0021-8693</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0245683">0245683</a></li></ul>