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<a id="X7E7EFDD1878F599D" name="X7E7EFDD1878F599D"></a>
<div class="ChapSects"><a href="chap4.html#X7E7EFDD1878F599D">4 T-groups, PT-groups, and PST-groups</a>
<div class="ContSect"> <a href="chap4.html#X7ABC289380DB85BE">4.1 "One" functions</a>

<div class="ContSSBlock">
 <a href="chap4.html#X7E15C5F778535A28">4.1-1 OneSubnormalNonNormalSubgroup</a>
 <a href="chap4.html#X7F5EE21F85A8D14F">4.1-2 OneSubnormalNonPermutableSubgroup</a>
 <a href="chap4.html#X7D7717657B9C7BEB">4.1-3 OneSubnormalNonSPermutableSubgroup</a>
 <a href="chap4.html#X7878496C869CB90C">4.1-4 OneSubnormalNonConjugatePermutableSubgroup</a>
 <a href="chap4.html#X7B23E3857CA178FE">4.1-5 OneSubnormalNonSNPermutableSubgroup</a>
</div></div>
<div class="ContSect"> <a href="chap4.html#X7B13DA0A800D8635">4.2 Group properties related to permutability</a>

<div class="ContSSBlock">
 <a href="chap4.html#X7930E5AD817BE4B3">4.2-1 IsTGroup</a>
 <a href="chap4.html#X7CF2634F84EC572B">4.2-2 IsPTGroup</a>
 <a href="chap4.html#X7BB4D5A782FF5B0E">4.2-3 IsPSTGroup</a>
 <a href="chap4.html#X8789176282320022">4.2-4 IsCPTGroup</a>
 <a href="chap4.html#X7B9EBEA07DBE246F">4.2-5 IsPSNTGroup</a>
</div></div>
</div>

<h3>4 T-groups, PT-groups, and PST-groups</h3>

This chapter explains the functions to check whether a given group is a T-group, a PT-group, or a PST-group.

Recall that a group G is:

<dl>
<dt>a T-group</dt>
<dd>when every subnormal subgroup of G is normal,

</dd>
<dt>a PT-group</dt>
<dd>when every subnormal subgroup of G is permutable,

</dd>
<dt>a PST-group</dt>
<dd>when every subnormal subgroup of G is S-permutable.

</dd>
</dl>
We also present functions to identify groups in other classes related to these ones.

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.

<a id="X7ABC289380DB85BE" name="X7ABC289380DB85BE"></a>

<h4>4.1 "One" functions</h4>

<a id="X7E15C5F778535A28" name="X7E15C5F778535A28"></a>

<h5>4.1-1 OneSubnormalNonNormalSubgroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OneSubnormalNonNormalSubgroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<code class="func">OneSubnormalNonNormalSubgroup</code> returns a subnormal subgroup of defect 2 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>.

A T-group is a group in which normality is transitive, that is, if H is a normal subgroup of K and K is a normal subgroup of G, then H is a normal subgroup of G. Finite T-groups are the groups in which every subnormal subgroup is normal.

This function tries to set the property <code class="func">IsTGroup</code> (<a href="chap4.html#X7930E5AD817BE4B3">4.2-1</a>) to <code class="keyw">true</code> or <code class="keyw">false</code> according to its result.

<div class="example"><pre>
gap> g:=SmallGroup(320,152);
<pc group of size 320 with 7 generators>
gap> x:=OneSubnormalNonNormalSubgroup(g);
Group([ f2, f3, f5, f7 ])
gap> IsNormal(g,x);
false
gap> IsSubnormal(g,x);
true
</pre></div>

<a id="X7F5EE21F85A8D14F" name="X7F5EE21F85A8D14F"></a>

<h5>4.1-2 OneSubnormalNonPermutableSubgroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OneSubnormalNonPermutableSubgroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<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>.

A group G is a PT-group when permutability is a transitive relation in G, that is, if H is a permutable subgroup of K and K is a permutable subgroup of G, then H is a permutable subgroup of G. This is equivalent in finite groups to affirming that every subnormal subgroup of G is permutable.

This function tries to set the property <code class="func">IsPTGroup</code> (<a href="chap4.html#X7CF2634F84EC572B">4.2-2</a>) to <code class="keyw">true</code> or <code class="keyw">false</code> according to its result.

Since this function checks all subnormal subgroups for permutability, it may take a long time if there are many subnormal subgroups.

<div class="example"><pre>
gap> g:=SmallGroup(320,152);
<pc group of size 320 with 7 generators>
gap> OneSubnormalNonPermutableSubgroup(g);
fail
gap> IsPTGroup(g);
true
gap> g:=SmallGroup(8,3);
<pc group of size 8 with 3 generators>
gap> OneSubnormalNonPermutableSubgroup(g);
Group([ f1*f3 ])
</pre></div>

<a id="X7D7717657B9C7BEB" name="X7D7717657B9C7BEB"></a>

<h5>4.1-3 OneSubnormalNonSPermutableSubgroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OneSubnormalNonSPermutableSubgroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<code class="func">OneSubnormalNonSPermutableSubgroup</code> returns a subnormal subgroup of defect 2 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>.

A group G is a PST-group when S-permutability (Sylow permutability) is a transitive relation in G, that is, if H is an S-permutable subgroup of K and K is an S-permutable subgroup of G, then H is an S-permutable subgroup of G. This is equivalent in finite groups to affirming that every subnormal subgroup of G is S-permutable. By a result of Ballester-Bolinches, Esteban-Romero, and Ragland <a href="chapBib.html#biBBallesterEstebanRagland07">[BBERR07]</a>, it is enough to check this last condition for all subnormal subgroups of defect 2.

This function tries to set the property <code class="func">IsPSTGroup</code> (<a href="chap4.html#X7BB4D5A782FF5B0E">4.2-3</a>) to <code class="keyw">true</code> or <code class="keyw">false</code> according to its result.

<div class="example"><pre>
gap> g:=AlternatingGroup(4);
Alt( [ 1 .. 4 ] )
gap> OneSubnormalNonSPermutableSubgroup(g);
Group([ (1,2)(3,4) ])
</pre></div>

<a id="X7878496C869CB90C" name="X7878496C869CB90C"></a>

<h5>4.1-4 OneSubnormalNonConjugatePermutableSubgroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OneSubnormalNonConjugatePermutableSubgroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
This function finds a subnormal subgroup H which does not permute with all its conjugates, if such a subgroup exist; otherwise, it returns <code class="keyw">fail</code>.

<div class="example"><pre>
gap> g:=AlternatingGroup(4);
Alt( [ 1 .. 4 ] )
gap> OneSubnormalNonConjugatePermutableSubgroup(g);
fail
gap> g:=DihedralGroup(16);
<pc group of size 16 with 4 generators>
gap> OneSubnormalNonConjugatePermutableSubgroup(g);
Group([ f1*f4 ])
gap> g:=SymmetricGroup(4);
Sym( [ 1 .. 4 ] )
gap> OneSubnormalNonConjugatePermutableSubgroup(g);
fail
gap> OneSubnormalNonPermutableSubgroup(g);
Group([ (1,2)(3,4) ])
</pre></div>

<a id="X7B23E3857CA178FE" name="X7B23E3857CA178FE"></a>

<h5>4.1-5 OneSubnormalNonSNPermutableSubgroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OneSubnormalNonSNPermutableSubgroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
This attribute returns a subnormal subgroup H of the soluble group G such that H 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">FORMAT: SystemNormalizer</a>) of the Format package.

<div class="example"><pre>
gap> g:=SymmetricGroup(4);
Sym( [ 1 .. 4 ] )
gap> OneSubnormalNonSNPermutableSubgroup(g);
Group([ (1,3)(2,4) ])
gap> g:=Group((1,2,3)(4,5,6),(1,2));
Group([ (1,2,3)(4,5,6), (1,2) ])
gap> OneSubnormalNonSNPermutableSubgroup(g);
fail
gap> OneSubnormalNonSPermutableSubgroup(g); 
Group([ (1,2,3)(4,6,5) ])
</pre></div>

<a id="X7B13DA0A800D8635" name="X7B13DA0A800D8635"></a>

<h4>4.2 Group properties related to permutability</h4>

The next function names correspond to properties.

<a id="X7930E5AD817BE4B3" name="X7930E5AD817BE4B3"></a>

<h5>4.2-1 IsTGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsTGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( property )</td></tr></table></div>
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.

T-groups are the groups in which normality is a transitive relation, that is, if H is a subgroup of K and K is a subgroup of G, then H is a subgroup of G. In the finite case, they are the groups in which every subnormal subgroup is normal.

For soluble groups, the algorithm checks that for every prime p dividing its order, G is p-nilpotent and has a Dedekind Sylow p-subgroup or G has an abelian Sylow p-subgroup P and every subgroup of P is normal in N_G(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.html#X7E01D1C47814DA3C">7.1-4</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">Reference: ChiefSeries</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 2 is normal with the help of the function <code class="func">OneSubnormalNonNormalSubgroup</code> (<a href="chap4.html#X7E15C5F778535A28">4.1-1</a>). The methods based on the ideas of <a href="chapBib.html#biBBallesterBeidlemanHeineken03-illinois">[BBBH03a]</a>, <a href="chapBib.html#biBBallesterBeidlemanHeineken03-commalg">[BBBH03b]</a>, and <a href="chapBib.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.

<div class="example"><pre>
gap> g:=SmallGroup(40,4);
<pc group of size 40 with 4 generators>
gap> IsTGroup(g);
true
gap> g:=SymmetricGroup(3);
Sym( [ 1 .. 3 ] )
gap> IsTGroup(g);
true
</pre></div>

<a id="X7CF2634F84EC572B" name="X7CF2634F84EC572B"></a>

<h5>4.2-2 IsPTGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsPTGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( property )</td></tr></table></div>
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.

For a soluble group G, the function checks whether for all primes p, G is p-nilpotent and has an Iwasawa Sylow p-subgroup or G has an abelian Sylow p-subgroup and it satisfies the property C_p (that is, every subgroup of a Sylow p-subgroup P of G is normal in the Sylow normaliser N_G(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.html#X7E01D1C47814DA3C">7.1-4</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">Reference: ChiefSeries</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.html#X7F5EE21F85A8D14F">4.1-2</a>) to check whether or not every subnormal subgroup is permutable. The methods based on the ideas of <a href="chapBib.html#biBBallesterBeidlemanHeineken03-illinois">[BBBH03a]</a>, <a href="chapBib.html#biBBallesterBeidlemanHeineken03-commalg">[BBBH03b]</a>, and <a href="chapBib.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.

<div class="example"><pre>
gap> g:=SmallGroup(1323,37);
<pc group of size 1323 with 5 generators>
gap> IsPTGroup(g);
true
gap> IsTGroup(g);
false
gap> OneSubnormalNonNormalSubgroup(g);
Group([ f2*f3, f4, f5 ])
</pre></div>

<a id="X7BB4D5A782FF5B0E" name="X7BB4D5A782FF5B0E"></a>

<h5>4.2-3 IsPSTGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsPSTGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( property )</td></tr></table></div>
This function returns true if the group <var class="Arg">G</var> is a PST-group, and false otherwise.

A finite group G is a PST-group if S-permutability (Sylow-permutability) is a transitive relation in G, that is, if H is S-permutable in K and K is S-permutable in G, then H is S-permutable in G. This is equivalent to affirming that every subnormal subgroup of G is S-permutable in G.

For a soluble group G, the function checks whether for all primes p, G is p-nilpotent, or G has an abelian Sylow p-subgroup and G satisfies the property C_p (that is, every subgroup of a Sylow p-subgroup P of G is normal in the Sylow normaliser N_G(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.html#X7E01D1C47814DA3C">7.1-4</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">Reference: ChiefSeries</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.html#X7D7717657B9C7BEB">4.1-3</a>) to check whether or not every subnormal subgroup of defect 2 is S-permutable. The methods based on the ideas of <a href="chapBib.html#biBBallesterBeidlemanHeineken03-illinois">[BBBH03a]</a>, <a href="chapBib.html#biBBallesterBeidlemanHeineken03-commalg">[BBBH03b]</a>, and <a href="chapBib.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.

<div class="example"><pre>
gap> g:=SmallGroup(24,6);
<pc group of size 24 with 4 generators>
gap> IsPSTGroup(g);
true
gap> IsPTGroup(g);
false
gap> OneSubnormalNonPermutableSubgroup(g);
Group([ f1*f3, f4 ])
gap> g:=SmallGroup(24,6);
<pc group of size 24 with 4 generators>
gap> IsPSTGroup(g);
true
gap> IsPTGroup(g);
false
gap> OneSubnormalNonPermutableSubgroup(g);
Group([ f1*f3, f4 ])
gap> OneSubgroupNotPermutingWith(g,last);
Group([ f1*f2 ])
</pre></div>

<a id="X8789176282320022" name="X8789176282320022"></a>

<h5>4.2-4 IsCPTGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsCPTGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( property )</td></tr></table></div>
This property returns true if every subnormal subgroup of <var class="Arg">G</var> permutes with all its conjugates, and false otherwise.

<div class="example"><pre>
gap> g:=SymmetricGroup(4);
Sym( [ 1 .. 4 ] )
gap> IsCPTGroup(g);
true
gap> IsPTGroup(g);
false
gap> IsPSTGroup(g);
false
</pre></div>

<a id="X7B9EBEA07DBE246F" name="X7B9EBEA07DBE246F"></a>

<h5>4.2-5 IsPSNTGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsPSNTGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( property )</td></tr></table></div>
This property takes the value <code class="keyw">true</code> if every subnormal subgroup of the soluble group G permutes with every system normaliser of G, and <code class="keyw">false</code> otherwise. If the function is applied to an insoluble group, it gives an error.

<div class="example"><pre>
gap> g:=Group((1,2,3)(4,5,6),(1,3));
Group([ (1,2,3)(4,5,6), (1,3) ])
gap> IsPSTGroup(g);
false
gap> IsPSNTGroup(g);
true
gap> IsCPTGroup(g);
true
gap> g:=SmallGroup(16,7);
<pc group of size 16 with 4 generators>
gap> IsPSTGroup(g);
true
gap> IsCPTGroup(g);
false
gap> g:=SymmetricGroup(4);
Sym( [ 1 .. 4 ] )
gap> IsPSNTGroup(g);
false
gap> IsCPTGroup(g);
true
</pre></div>

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