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<div class="ChapSects"><a href="chap4_mj.html#X7E7EFDD1878F599D">4 <span class="Heading">T-groups, PT-groups, and PST-groups</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X7ABC289380DB85BE">4.1 <span class="Heading">"One" functions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7E15C5F778535A28">4.1-1 OneSubnormalNonNormalSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7F5EE21F85A8D14F">4.1-2 OneSubnormalNonPermutableSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7D7717657B9C7BEB">4.1-3 OneSubnormalNonSPermutableSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7878496C869CB90C">4.1-4 OneSubnormalNonConjugatePermutableSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7B23E3857CA178FE">4.1-5 OneSubnormalNonSNPermutableSubgroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X7B13DA0A800D8635">4.2 <span class="Heading">Group properties related to permutability</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7930E5AD817BE4B3">4.2-1 IsTGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7CF2634F84EC572B">4.2-2 IsPTGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7BB4D5A782FF5B0E">4.2-3 IsPSTGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X8789176282320022">4.2-4 IsCPTGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7B9EBEA07DBE246F">4.2-5 IsPSNTGroup</a></span>
</div></div>
</div>
<h3>4 <span class="Heading">T-groups, PT-groups, and PST-groups</span></h3>
<p>This chapter explains the functions to check whether a given group is a T-group, a PT-group, or a PST-group.</p>
<p>Recall that a group <span class="SimpleMath">\(G\)</span> is:</p>
<dl>
<dt><strong class="Mark">a T-group</strong></dt>
<dd><p>when every subnormal subgroup of <span class="SimpleMath">\(G\)</span> is normal,</p>
</dd>
<dt><strong class="Mark">a PT-group</strong></dt>
<dd><p>when every subnormal subgroup of <span class="SimpleMath">\(G\)</span> is permutable,</p>
</dd>
<dt><strong class="Mark">a PST-group</strong></dt>
<dd><p>when every subnormal subgroup of <span class="SimpleMath">\(G\)</span> is S-permutable.</p>
</dd>
</dl>
<p>We also present functions to identify groups in other classes related to these ones.</p>
<p>The "One" functions are defined to provide examples of subgroups or elements showing that a group theoretical property for a group or for a subgroup is false.</p>
<p><a id="X7ABC289380DB85BE" name="X7ABC289380DB85BE"></a></p>
<h4>4.1 <span class="Heading">"One" functions</span></h4>
<p><a id="X7E15C5F778535A28" name="X7E15C5F778535A28"></a></p>
<h5>4.1-1 OneSubnormalNonNormalSubgroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; OneSubnormalNonNormalSubgroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p><code class="func">OneSubnormalNonNormalSubgroup</code> returns a subnormal subgroup of defect <span class="SimpleMath">\(2\)</span> which is not normal in the group <var class="Arg">G</var>, if such a subgroup exists. If such a subgroup does not exist because the group is a T-group, it returns <code class="keyw">fail</code>.</p>
<p>A T-group is a group in which normality is transitive, that is, if <span class="SimpleMath">\(H\)</span> is a normal subgroup of <span class="SimpleMath">\(K\)</span> and <span class="SimpleMath">\(K\)</span> is a normal subgroup of <span class="SimpleMath">\(G\)</span>, then <span class="SimpleMath">\(H\)</span> is a normal subgroup of <span class="SimpleMath">\(G\)</span>. Finite T-groups are the groups in which every subnormal subgroup is normal.</p>
<p>This function tries to set the property <code class="func">IsTGroup</code> (<a href="chap4_mj.html#X7930E5AD817BE4B3"><span class="RefLink">4.2-1</span></a>) to <code class="keyw">true</code> or <code class="keyw">false</code> according to its result.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=SmallGroup(320,152);</span>
<pc group of size 320 with 7 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">x:=OneSubnormalNonNormalSubgroup(g);</span>
Group([ f2, f3, f5, f7 ])
<span class="GAPprompt">gap></span> <span class="GAPinput">IsNormal(g,x);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSubnormal(g,x);</span>
true
</pre></div>
<p><a id="X7F5EE21F85A8D14F" name="X7F5EE21F85A8D14F"></a></p>
<h5>4.1-2 OneSubnormalNonPermutableSubgroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; OneSubnormalNonPermutableSubgroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p><code class="func">OneSubnormalNonPermutableSubgroup</code> returns a subnormal subgroup which is not permutable in the group <var class="Arg">G</var>, if such a subgroup exists. If such a subgroup does not exist because the group is a PT-group, it returns <code class="keyw">fail</code>.</p>
<p>A group <span class="SimpleMath">\(G\)</span> is a PT-group when permutability is a transitive relation in <span class="SimpleMath">\(G\)</span>, that is, if <span class="SimpleMath">\(H\)</span> is a permutable subgroup of <span class="SimpleMath">\(K\)</span> and <span class="SimpleMath">\(K\)</span> is a permutable subgroup of <span class="SimpleMath">\(G\)</span>, then <span class="SimpleMath">\(H\)</span> is a permutable subgroup of <span class="SimpleMath">\(G\)</span>. This is equivalent in finite groups to affirming that every subnormal subgroup of <span class="SimpleMath">\(G\)</span> is permutable.</p>
<p>This function tries to set the property <code class="func">IsPTGroup</code> (<a href="chap4_mj.html#X7CF2634F84EC572B"><span class="RefLink">4.2-2</span></a>) to <code class="keyw">true</code> or <code class="keyw">false</code> according to its result.</p>
<p>Since this function checks all subnormal subgroups for permutability, it may take a long time if there are many subnormal subgroups.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=SmallGroup(320,152);</span>
<pc group of size 320 with 7 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">OneSubnormalNonPermutableSubgroup(g);</span>
fail
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPTGroup(g);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=SmallGroup(8,3);</span>
<pc group of size 8 with 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">OneSubnormalNonPermutableSubgroup(g);</span>
Group([ f1*f3 ])
</pre></div>
<p><a id="X7D7717657B9C7BEB" name="X7D7717657B9C7BEB"></a></p>
<h5>4.1-3 OneSubnormalNonSPermutableSubgroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; OneSubnormalNonSPermutableSubgroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p><code class="func">OneSubnormalNonSPermutableSubgroup</code> returns a subnormal subgroup of defect <span class="SimpleMath">\(2\)</span> which is not S-permutable in <var class="Arg">G</var>, if such a subgroup exists. If such a subgroup does not exist because the group is a PST-group, it returns <code class="keyw">fail</code>.</p>
<p>A group <span class="SimpleMath">\(G\)</span> is a PST-group when S-permutability (Sylow permutability) is a transitive relation in <span class="SimpleMath">\(G\)</span>, that is, if <span class="SimpleMath">\(H\)</span> is an S-permutable subgroup of <span class="SimpleMath">\(K\)</span> and <span class="SimpleMath">\(K\)</span> is an S-permutable subgroup of <span class="SimpleMath">\(G\)</span>, then <span class="SimpleMath">\(H\)</span> is an S-permutable subgroup of <span class="SimpleMath">\(G\)</span>. This is equivalent in finite groups to affirming that every subnormal subgroup of <span class="SimpleMath">\(G\)</span> is S-permutable. By a result of Ballester-Bolinches, Esteban-Romero, and Ragland <a href="chapBib_mj.html#biBBallesterEstebanRagland07">[BBERR07]</a>, it is enough to check this last condition for all subnormal subgroups of defect <span class="SimpleMath">\(2\)</span>.</p>
<p>This function tries to set the property <code class="func">IsPSTGroup</code> (<a href="chap4_mj.html#X7BB4D5A782FF5B0E"><span class="RefLink">4.2-3</span></a>) to <code class="keyw">true</code> or <code class="keyw">false</code> according to its result.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=AlternatingGroup(4);</span>
Alt( [ 1 .. 4 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">OneSubnormalNonSPermutableSubgroup(g);</span>
Group([ (1,2)(3,4) ])
</pre></div>
<p><a id="X7878496C869CB90C" name="X7878496C869CB90C"></a></p>
<h5>4.1-4 OneSubnormalNonConjugatePermutableSubgroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; OneSubnormalNonConjugatePermutableSubgroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>This function finds a subnormal subgroup <span class="SimpleMath">\(H\)</span> which does not permute with all its conjugates, if such a subgroup exist; otherwise, it returns <code class="keyw">fail</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=AlternatingGroup(4);</span>
Alt( [ 1 .. 4 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">OneSubnormalNonConjugatePermutableSubgroup(g);</span>
fail
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=DihedralGroup(16);</span>
<pc group of size 16 with 4 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">OneSubnormalNonConjugatePermutableSubgroup(g);</span>
Group([ f1*f4 ])
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=SymmetricGroup(4);</span>
Sym( [ 1 .. 4 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">OneSubnormalNonConjugatePermutableSubgroup(g);</span>
fail
<span class="GAPprompt">gap></span> <span class="GAPinput">OneSubnormalNonPermutableSubgroup(g);</span>
Group([ (1,2)(3,4) ])
</pre></div>
<p><a id="X7B23E3857CA178FE" name="X7B23E3857CA178FE"></a></p>
<h5>4.1-5 OneSubnormalNonSNPermutableSubgroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; OneSubnormalNonSNPermutableSubgroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>This attribute returns a subnormal subgroup <span class="SimpleMath">\(H\)</span> of the soluble group <span class="SimpleMath">\(G\)</span> such that <span class="SimpleMath">\(H\)</span> does not permute with a system normaliser if such a subgroup exists; otherwise, it returns <code class="keyw">fail</code>. This system normaliser is obtained with the function <code class="func">SystemNormalizer</code> (<a href="../../../pkg/format/htm/CHAP004.htm"><span class="RefLink">FORMAT: SystemNormalizer</span></a>) of the <strong class="pkg">Format</strong> package.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=SymmetricGroup(4);</span>
Sym( [ 1 .. 4 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">OneSubnormalNonSNPermutableSubgroup(g);</span>
Group([ (1,3)(2,4) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=Group((1,2,3)(4,5,6),(1,2));</span>
Group([ (1,2,3)(4,5,6), (1,2) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">OneSubnormalNonSNPermutableSubgroup(g);</span>
fail
<span class="GAPprompt">gap></span> <span class="GAPinput">OneSubnormalNonSPermutableSubgroup(g); </span>
Group([ (1,2,3)(4,6,5) ])
</pre></div>
<p><a id="X7B13DA0A800D8635" name="X7B13DA0A800D8635"></a></p>
<h4>4.2 <span class="Heading">Group properties related to permutability</span></h4>
<p>The next function names correspond to properties.</p>
<p><a id="X7930E5AD817BE4B3" name="X7930E5AD817BE4B3"></a></p>
<h5>4.2-1 IsTGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; IsTGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>This function returns <code class="keyw">true</code> if <var class="Arg">G</var> is a T-group, and <code class="keyw">false</code> otherwise.</p>
<p>T-groups are the groups in which normality is a transitive relation, that is, if <span class="SimpleMath">\(H\)</span> is a subgroup of <span class="SimpleMath">\(K\)</span> and <span class="SimpleMath">\(K\)</span> is a subgroup of <span class="SimpleMath">\(G\)</span>, then <span class="SimpleMath">\(H\)</span> is a subgroup of <span class="SimpleMath">\(G\)</span>. In the finite case, they are the groups in which every subnormal subgroup is normal.</p>
<p>For soluble groups, the algorithm checks that for every prime <span class="SimpleMath">\(p\)</span> dividing its order, <span class="SimpleMath">\(G\)</span> is <span class="SimpleMath">\(p\)</span>-nilpotent and has a Dedekind Sylow <span class="SimpleMath">\(p\)</span>-subgroup or <span class="SimpleMath">\(G\)</span> has an abelian Sylow <span class="SimpleMath">\(p\)</span>-subgroup <span class="SimpleMath">\(P\)</span> and every subgroup of <span class="SimpleMath">\(P\)</span> is normal in <span class="SimpleMath">\({\rm N}_G(P)\)</span>.</p>
<p>For insoluble groups, the function checks whether the group is an SC-group with the function <code class="func">IsSCGroup</code> (<a href="chap7_mj.html#X7E01D1C47814DA3C"><span class="RefLink">7.1-4</span></a>), because PT-groups are SC-groups. Since the methods for insoluble groups depend on the computation of a chief series with the function <code class="func">ChiefSeries</code> (<a href="../../../doc/ref/chap39_mj.html#X7BDD116F7833800F"><span class="RefLink">Reference: ChiefSeries</span></a>), they might not be available if the group is not given as a permutation group. Then it is checked that every subnormal subgroup of defect <span class="SimpleMath">\(2\)</span> is normal with the help of the function <code class="func">OneSubnormalNonNormalSubgroup</code> (<a href="chap4_mj.html#X7E15C5F778535A28"><span class="RefLink">4.1-1</span></a>). The methods based on the ideas of <a href="chapBib_mj.html#biBBallesterBeidlemanHeineken03-illinois">[BBBH03a]</a>, <a href="chapBib_mj.html#biBBallesterBeidlemanHeineken03-commalg">[BBBH03b]</a>, and <a href="chapBib_mj.html#biBBeidlemanHeineken03-jgt">[BH03]</a> have not been implemented so far because they require the computation of quotients by all normal subgroups, which could be a time-consuming task.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=SmallGroup(40,4);</span>
<pc group of size 40 with 4 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsTGroup(g);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=SymmetricGroup(3);</span>
Sym( [ 1 .. 3 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">IsTGroup(g);</span>
true
</pre></div>
<p><a id="X7CF2634F84EC572B" name="X7CF2634F84EC572B"></a></p>
<h5>4.2-2 IsPTGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; IsPTGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>This property takes the value <code class="keyw">true</code> if <var class="Arg">G</var> is a PT-group, and the value <code class="keyw">false</code> otherwise.</p>
<p>For a soluble group <span class="SimpleMath">\(G\)</span>, the function checks whether for all primes <span class="SimpleMath">\(p\)</span>, <span class="SimpleMath">\(G\)</span> is <span class="SimpleMath">\(p\)</span>-nilpotent and has an Iwasawa Sylow <span class="SimpleMath">\(p\)</span>-subgroup or <span class="SimpleMath">\(G\)</span> has an abelian Sylow <span class="SimpleMath">\(p\)</span>-subgroup and it satisfies the property <span class="SimpleMath">\(C_p\)</span> (that is, every subgroup of a Sylow <span class="SimpleMath">\(p\)</span>-subgroup <span class="SimpleMath">\(P\)</span> of <span class="SimpleMath">\(G\)</span> is normal in the Sylow normaliser <span class="SimpleMath">\({\rm N}_G(P)\)</span>).</p>
<p>For insoluble groups, the function checks that the group is an SC-group with the function <code class="func">IsSCGroup</code> (<a href="chap7_mj.html#X7E01D1C47814DA3C"><span class="RefLink">7.1-4</span></a>), because PT-groups are SC-groups. Since the methods for insoluble groups depend on the computation of a chief series with the function <code class="func">ChiefSeries</code> (<a href="../../../doc/ref/chap39_mj.html#X7BDD116F7833800F"><span class="RefLink">Reference: ChiefSeries</span></a>), they might not be available if the group is not given as a permutation group. Then it uses the function <code class="func">OneSubnormalNonPermutableSubgroup</code> (<a href="chap4_mj.html#X7F5EE21F85A8D14F"><span class="RefLink">4.1-2</span></a>) to check whether or not every subnormal subgroup is permutable. The methods based on the ideas of <a href="chapBib_mj.html#biBBallesterBeidlemanHeineken03-illinois">[BBBH03a]</a>, <a href="chapBib_mj.html#biBBallesterBeidlemanHeineken03-commalg">[BBBH03b]</a>, and <a href="chapBib_mj.html#biBBeidlemanHeineken03-jgt">[BH03]</a> have not been implemented so far because they require the computation of quotients by all normal subgroups, which could be a time-consuming task.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=SmallGroup(1323,37);</span>
<pc group of size 1323 with 5 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPTGroup(g);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsTGroup(g);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">OneSubnormalNonNormalSubgroup(g);</span>
Group([ f2*f3, f4, f5 ])
</pre></div>
<p><a id="X7BB4D5A782FF5B0E" name="X7BB4D5A782FF5B0E"></a></p>
<h5>4.2-3 IsPSTGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; IsPSTGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>This function returns true if the group <var class="Arg">G</var> is a PST-group, and false otherwise.</p>
<p>A finite group <span class="SimpleMath">\(G\)</span> is a PST-group if S-permutability (Sylow-permutability) is a transitive relation in <span class="SimpleMath">\(G\)</span>, that is, if <span class="SimpleMath">\(H\)</span> is S-permutable in <span class="SimpleMath">\(K\)</span> and <span class="SimpleMath">\(K\)</span> is S-permutable in <span class="SimpleMath">\(G\)</span>, then <span class="SimpleMath">\(H\)</span> is S-permutable in <span class="SimpleMath">\(G\)</span>. This is equivalent to affirming that every subnormal subgroup of <span class="SimpleMath">\(G\)</span> is S-permutable in <span class="SimpleMath">\(G\)</span>.</p>
<p>For a soluble group <span class="SimpleMath">\(G\)</span>, the function checks whether for all primes <span class="SimpleMath">\(p\)</span>, <span class="SimpleMath">\(G\)</span> is <span class="SimpleMath">\(p\)</span>-nilpotent, or <span class="SimpleMath">\(G\)</span> has an abelian Sylow <span class="SimpleMath">\(p\)</span>-subgroup and <span class="SimpleMath">\(G\)</span> satisfies the property <span class="SimpleMath">\(C_p\)</span> (that is, every subgroup of a Sylow <span class="SimpleMath">\(p\)</span>-subgroup <span class="SimpleMath">\(P\)</span> of <span class="SimpleMath">\(G\)</span> is normal in the Sylow normaliser <span class="SimpleMath">\({\rm N}_G(P)\)</span>)</p>
<p>For insoluble groups, the function checks whether the group is an SC-group with the function <code class="func">IsSCGroup</code> (<a href="chap7_mj.html#X7E01D1C47814DA3C"><span class="RefLink">7.1-4</span></a>), because PST-groups are SC-groups. Since the methods for insoluble groups depend on the computation of a chief series with the function <code class="func">ChiefSeries</code> (<a href="../../../doc/ref/chap39_mj.html#X7BDD116F7833800F"><span class="RefLink">Reference: ChiefSeries</span></a>), they might not be available if the group is not given as a permutation group. Then it uses the function <code class="func">OneSubnormalNonSPermutableSubgroup</code> (<a href="chap4_mj.html#X7D7717657B9C7BEB"><span class="RefLink">4.1-3</span></a>) to check whether or not every subnormal subgroup of defect <span class="SimpleMath">\(2\)</span> is S-permutable. The methods based on the ideas of <a href="chapBib_mj.html#biBBallesterBeidlemanHeineken03-illinois">[BBBH03a]</a>, <a href="chapBib_mj.html#biBBallesterBeidlemanHeineken03-commalg">[BBBH03b]</a>, and <a href="chapBib_mj.html#biBBeidlemanHeineken03-jgt">[BH03]</a> have not been implemented so far because they require the computation of quotients by all normal subgroups, which could be a time-consuming task.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=SmallGroup(24,6);</span>
<pc group of size 24 with 4 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPSTGroup(g);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPTGroup(g);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">OneSubnormalNonPermutableSubgroup(g);</span>
Group([ f1*f3, f4 ])
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=SmallGroup(24,6);</span>
<pc group of size 24 with 4 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPSTGroup(g);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPTGroup(g);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">OneSubnormalNonPermutableSubgroup(g);</span>
Group([ f1*f3, f4 ])
<span class="GAPprompt">gap></span> <span class="GAPinput">OneSubgroupNotPermutingWith(g,last);</span>
Group([ f1*f2 ])
</pre></div>
<p><a id="X8789176282320022" name="X8789176282320022"></a></p>
<h5>4.2-4 IsCPTGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; IsCPTGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>This property returns true if every subnormal subgroup of <var class="Arg">G</var> permutes with all its conjugates, and false otherwise.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=SymmetricGroup(4);</span>
Sym( [ 1 .. 4 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">IsCPTGroup(g);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPTGroup(g);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPSTGroup(g);</span>
false
</pre></div>
<p><a id="X7B9EBEA07DBE246F" name="X7B9EBEA07DBE246F"></a></p>
<h5>4.2-5 IsPSNTGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">an style='color: green'>8227; IsPSNTGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>This property takes the value <code class="keyw">true</code> if every subnormal subgroup of the soluble group <span class="SimpleMath">\(G\)</span> permutes with every system normaliser of <span class="SimpleMath">\(G\)</span>, and <code class="keyw">false</code> otherwise. If the function is applied to an insoluble group, it gives an error.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=Group((1,2,3)(4,5,6),(1,3));</span>
Group([ (1,2,3)(4,5,6), (1,3) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPSTGroup(g);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPSNTGroup(g);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsCPTGroup(g);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=SmallGroup(16,7);</span>
<pc group of size 16 with 4 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPSTGroup(g);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsCPTGroup(g);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=SymmetricGroup(4);</span>
Sym( [ 1 .. 4 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPSNTGroup(g);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsCPTGroup(g);</span>
true
</pre></div>
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