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<h1>5 Primitive soluble groups</h1>

<H3>Sections</H3>
<oL>
<li> <A HREF="CHAP005.htm#SECT001">Converting between irreducible soluble matrix groups and primitive soluble groups</a>
<li> <A HREF="CHAP005.htm#SECT002">Finding primitive pc groups with given properties</a>
<li> <A HREF="CHAP005.htm#SECT003">Finding primitive soluble permutation groups with given properties</a>
<li> <A HREF="CHAP005.htm#SECT004">Recognising primitive soluble groups</a>
<li> <A HREF="CHAP005.htm#SECT005">Obsolete functions</a>
</ol>

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<a name = "I1"></a>

<a name = "I2"></a>

<a name = "I3"></a>

An abstract finite group <var>G</var> is called primitiveif it has a maximal
subgroup <var>M</var> with trivial core. Note that the permutation action of <var>G</var> on the
cosets of <var>M</var> is faithful and primitive. Conversely, if <var>G</var> is a primitive
permutation group, then a point stabilizer <var>M</var> of <var>G</var> is a maximal
subgroup with trivial core. However, a permutation group which is primitive
as an abstract group need not be primitive as a permutation group.

Now assume that <var>G</var> is primitive and soluble. Then there exists a unique conjugacy class of
such maximal subgroups <var>M</var>; the index of <var>M</var> in <var>G</var> is called the degreeof <var>G</var>.
Moreover, <var><var>M</var></var>
complements the socle <var>N</var> of <var><var>G</var></var>. THe socle <var>N</var> coincides with the Fitting
subgroup of <var>G</var>; it is the unique minimal normal subgroup <var>N</var> of <var>G</var>. Therefore,
the index of <var>M</var> in <var>G</var> is a prime power, <var>pn</var>, say.
Regarding <var>N</var> as a <var>Fp</var>-vector space, <var>M</var> acts as an irreducible subgroup of <var>GL(n,p)</var> on <var>N</var>.
Conversely, if <var>M</var> is an irreducible soluble subgroup of <var>GL(n,p)</var>, and <var>V = Fpn</var>, then the
split extension of <var>V</var> by <var>M</var> is a primitive soluble group.
This establishes a well known
bijection between the isomorphism types (or, equivalently, the
<var>Sym(pn)</var>-conjugacy classes) of primitive soluble permutation groups of degree
<var><var>p</var><n></var> and the conjugacy classes of irreducible soluble subgroups of <var>GL(n, p)</var>.

The IRREDSOL package provides functions for translating between primitive soluble
groups and irreducible soluble matrix groups, which are
described in Section <a href="CHAP005.htm#SECT001">Converting between irreducible soluble matrix groups and primitive soluble groups</a>. Moreover, there are functions for finding
primitive soluble groups with given properties, see Sections <a href="CHAP005.htm#SECT002">Finding primitive pc groups with given properties</a> and <a href="CHAP005.htm#SECT003">Finding primitive soluble permutation groups with given properties</a>.


<h2><a name="SECT001">5.1 Converting between irreducible soluble matrix groups and primitive soluble groups</a></h2>

<a name = "SSEC001.1"></a>
<li><code>PrimitivePcGroup(</code><var>n</var><code>,</code><var>p</var><code>,</code><var>d</var><code>,</code><var>k</var><code>) F</code>
<a name = "SSEC001.1"></a>
<li><code>PrimitiveSolublePermGroup(</code><var>n</var><code>,</code><var>p</var><code>,</code><var>d</var><code>,</code><var>k</var><code>) F</code>
<a name = "SSEC001.1"></a>
<li><code>PrimitiveSolvablePermGroup(</code><var>n</var><code>,</code><var>p</var><code>,</code><var>d</var><code>,</code><var>k</var><code>) F</code>

These functions return the primitive soluble pc group resp. primitive soluble
permutation group obtainewd as the natural split extension of <var>V = Fpn</var> by
<code>IrreducibleSolubleMatrixGroup</code>(<var>n</var>,<var>p</var>,<var>d</var>,<var>k</var>). Here, <var>n</var> is a positive
integer, <var>p</var> is a prime, <var>d</var> divides <var>n</var> and <var>k</var> occurs in the list
<code>IndicesIrreducibleSolubleMatrixGroups</code>(<var>n</var>,<var>p</var>,<var>d</var>)
(see <a href="CHAP002.htm#SSEC002.2">IndicesIrreducibleSolubleMatrixGroups</a>).

As long as the relevant group data is not unloaded manually
(see <a href="CHAP002.htm#SSEC004.3">UnloadAbsolutelyIrreducibleSolubleGroupData</a>), the functions <code>PrimitivePcGroup</code>
and <code>PrimitiveSolublePermGroup</code> will return the same group when called multiple times
with the same arguments.

<a name = "SSEC001.2"></a>
<li><code>PrimitivePcGroupIrreducibleMatrixGroup(</code><var>G</var><code>) F</code>
<a name = "SSEC001.2"></a>
<li><code>PrimitivePcGroupIrreducibleMatrixGroupNC(</code><var>G</var><code>) F</code>

For a given irreducible soluble matrix group <var>G</var> over a prime field, this function
returns a primitive pc group <var>H</var> which is the split extension of <var>G</var> with its natural
underlying vector space <var>V</var>. The <code>NC</code> version does not check whether <var>G</var> is over a prime field,
or whether <var>G</var> is irreducible. The group <var>H</var> has an attribute <code>Socle</code> (see <a href="../../../doc/ref/chap39.html#X81F647FA83D8854F">Socle</a>), corresponding to <var>V</var>. If the package CRISP is loaded,
then the attribute <code>SocleComplement</code> (see CRISP reference manual <a href="../../crisp/htm/CHAP004.htm#SSEC003.2">crisp:SocleComplement</a>) is set to a subgroup of
<var>H</var> isomorphic with <var>G</var>.

<pre>
gap> PrimitivePcGroupIrreducibleMatrixGroup(
> IrreducibleSolubleMatrixGroup(4,2,2,3));
<pc group of size 160 with 6 generators>
</pre>

<a name = "SSEC001.3"></a>
<li><code>PrimitivePermGroupIrreducibleMatrixGroup(</code><var>G</var><code>) F</code>
<a name = "SSEC001.3"></a>
<li><code>PrimitivePermGroupIrreducibleMatrixGroupNC(</code><var>G</var><code>) F</code>

For a given irreducible soluble matrix group <var>G</var> over a prime field, this function
returns a primitive permutation group <var>H</var>, representing the affine action of <var>G</var> on its natural
vector space <var>V</var>. The <code>NC</code> version does not check whether <var>G</var> is over a prime field,
or whether <var>G</var> is irreducible. The group <var>H</var> has an attribute <code>Socle</code> (see <a href="../../../doc/ref/chap39.html#X81F647FA83D8854F">Socle</a>), corresponding to <var>V</var>. If the package CRISP is loaded,
then the attribute <code>SocleComplement</code> (see <a href="../../crisp/htm/CHAP004.htm#SSEC003.2">crisp:SocleComplement</a> in the CRISP manual) is set to a subgroup of
<var>H</var> isomorphic with <var>G</var>.

<pre>
gap> PrimitivePermGroupIrreducibleMatrixGroup(
> IrreducibleSolubleMatrixGroup(4,2,2,3));
<permutation group of size 160 with 6 generators>
</pre>

<a name = "SSEC001.4"></a>
<li><code>IrreducibleMatrixGroupPrimitiveSolubleGroup(</code><var>G</var><code>) F</code>
<a name = "SSEC001.4"></a>
<li><code>IrreducibleMatrixGroupPrimitiveSolvableGroupNC(</code><var>G</var><code>) F</code>

For a given primitive soluble group <var>G</var>, this function returns a matrix group
obtained from the conjugation
action of <var>G</var> on its unique minimal normal subgroup <var>N</var>, regarded as a vector
space over <var>Fp</var>, where <var>p</var> is the exponent of <var>N</var>.
The <var>Fp</var>-basis of <var>N</var> is chosen arbitrarily, so that the matrix group returned
is unique only up to conjugacy in the relevant <var>GL(n, p)</var>. The NC version does
not check whether <var>G</var> is primitive and soluble.

<pre>
gap> IrreducibleMatrixGroupPrimitiveSolubleGroup(SymmetricGroup(4));
Group([ <an immutable 2x2 matrix over GF2>,
 <an immutable 2x2 matrix over GF2> ])
</pre>


<h2><a name="SECT002">5.2 Finding primitive pc groups with given properties</a></h2>

<a name = "I4"></a>

<a name = "I5"></a>

<a name = "SSEC002.1"></a>
<li><code>AllPrimitivePcGroups(</code><var>func_1</var><code>, </code><var>arg_1</var><code>, </code><var>func_2</var><code>, </code><var>arg_2</var><code>, ...) F</code>

This function returns a list of all primitive soluble pc
groups <var>G</var> in the IRREDSOL library for which the return value of <var><var>funci</var>(G)</var> lies
in <var>arg_i</var>. The arguments <var>func_1</var>, <var>func_2</var>, ...,
must be GAP functions which take a pc group as their only argument and return a
value, and <var>arg_1</var>, <var>arg_2</var>,
..., must be lists. If <var>arg_i</var> is not a list, <var>arg_i</var> is replaced by the list
<code></code>[<var>arg_i</var><code>]</code>. One of the functions must be <code>Degree</code> or one of its
equivalents, see below.

The following functions <var>func_i</var> are handled particularly efficiently.

<ul>
<li> <code>Degree</code>, <code>NrMovedPoints</code>, <code>LargestMovedPoint</code>
<li> <code>Order</code>, <code>Size</code>
</ul>

Note that there is also a function <code>IteratorPrimitivePcGroups</code> (see
<a href="CHAP005.htm#SSEC002.3">IteratorPrimitivePcGroups</a>) which allows one to run through the list produced by
<code>AllPrimitivePcGroups</code> without having to store all the groups in the list
simultaneously.

<pre>
gap> AllPrimitivePcGroups(Degree, [1..255], Order, [168]);
[ <pc group of size 168 with 5 generators> ]
</pre>

<a name = "SSEC002.2"></a>
<li><code>OnePrimitivePcGroup(</code><var>func_1</var><code>, </code><var>arg_1</var><code>, </code><var>func_2</var><code>, </code><var>arg_2</var><code>, ...) F</code>

This function returns one primitive soluble pc
group <var>G</var> in the IRREDSOL library for which the return value of <var><var>funci</var>(G)</var> lies in
<var>arg_i</var>, or <code>fail</code> if no such group exists. The arguments <var>func_1</var>, <var>func_2</var>, ...,
must be GAP functions which take a pc group as their only argument and return a
value, and <var>arg_1</var>, <var>arg_2</var>,
..., must be lists. If <var>arg_i</var> is not a list, <var>arg_i</var> is replaced by the list
<code>[</code><var>arg_i</var><code>]</code>. One of the functions must be <code>Degree</code> or one of its, equivalents, <code>NrMovedPoints</code> or <code>LargestMovedPoint</code>.

For a list of functions which are handled particularly efficiently, see
<a href="CHAP005.htm#SSEC002.1">AllPrimitivePcGroups</a>.

<pre>
gap> OnePrimitivePcGroup(Degree, [256], Order, [256*255]);
<pc group of size 65280 with 11 generators>
</pre>

<a name = "SSEC002.3"></a>
<li><code>IteratorPrimitivePcGroups(</code><var>func_1</var><code>, </code><var>arg_1</var><code>, </code><var>func_2</var><code>, </code><var>arg_2</var><code>, ...) F</code>

This function returns an iterator which runs through the list of all primitive soluble
pc groups <var>G</var> in the IRREDSOL library such that
<var><var>funci</var>(G)</var> lies in <var>arg_i</var>. The arguments <var>func_1</var>, <var>func_2</var>, ...,
must be GAP functions taking a pc group as their only argument and returning
a value, and <var>arg_1</var>, <var>arg_2</var>, ...,
must be lists. If <var>arg_i</var> is not a list, <var>arg_i</var> is replaced by the list <code>[</code><var>arg_i</var><code>]</code>.
One of the functions must be <code>Degree</code> or one of its, equivalents, <code>NrMovedPoints</code>
or <code>LargestMovedPoint</code>.
For a list of functions which are handled particularly efficiently, see
<a href="CHAP005.htm#SSEC002.1">AllPrimitivePcGroups</a>.

Using

<code>IteratorPrimitivePcGroups</code>(<var>func_1</var>, <var>arg_1</var>, <var>func_2</var>, <var>arg_2</var>, ...))

is functionally equivalent to

<code>Iterator</code>(<code>AllPrimitivePcGroups</code>(<var>func_1</var>, <var>arg_1</var>, <var>func_2</var>, <var>arg_2</var>, ...))

(see <a href="../../../doc/ref/chap30.html#X85A3F00985453F95">Iterators</a> for details) but does not
compute all relevant pc groups at the same time.
This may save some memory.


<h2><a name="SECT003">5.3 Finding primitive soluble permutation groups with given properties</a></h2>

<a name = "I6"></a>

<a name = "I7"></a>

<a name = "SSEC003.1"></a>
<li><code>AllPrimitiveSolublePermGroups(</code><var>func_1</var><code>, </code><var>arg_1</var><code>, </code><var>func_2</var><code>, </code><var>arg_2</var><code>, ...) F</code>
<a name = "SSEC003.1"></a>
<li><code>AllPrimitiveSolvablePermGroups(</code><var>func_1</var><code>, </code><var>arg_1</var><code>, </code><var>func_2</var><code>, </code><var>arg_2</var><code>, ...) F</code>

This function returns a list of all primitive soluble permutation
groups <var>G</var> corresponding to irreducible matrix groups in the IRREDSOL library
for which the return value of <var><var>funci</var>(G)</var> lies
in <var>arg_i</var>. The arguments <var>func_1</var>, <var>func_2</var>, ...,
must be GAP functions which take a permutation group as their only argument and return a
value, and <var>arg_1</var>, <var>arg_2</var>,
..., must be lists. If <var>arg_i</var> is not a list, <var>arg_i</var> is replaced by the list
<code>[</code><var>arg_i</var><code>]</code>. One of the functions must be <code>Degree</code> or one of its
equivalents, see below.

The following functions <var>func_i</var> are handled particularly efficiently.

<ul>

<li> <code>Degree</code>, <code>NrMovedPoints</code>, <code>LargestMovedPoint</code>
<li> <code>Order</code>, <code>Size</code>
</ul>

Note that there is also a function <code>IteratorPrimitivePermGroups</code> (see
<a href="CHAP005.htm#SSEC003.3">IteratorPrimitivePermGroups</a>) which allows one to run through the list produced by
<code>AllPrimitivePcGroups</code> without having to store all of the groups
simultaneously.

<pre>
gap> AllPrimitiveSolublePermGroups(Degree, [1..100], Order, [72]);
[ Group([ (1,4,7)(2,5,8)(3,6,9), (1,2,3)(4,5,6)(7,8,9), (2,4)(3,7)(6,8),
 (2,3)(5,6)(8,9), (4,7)(5,8)(6,9) ]),
 Group([ (1,4,7)(2,5,8)(3,6,9), (1,2,3)(4,5,6)(7,8,9), (2,5,3,9)(4,8,7,6),
 (2,7,3,4)(5,8,9,6), (2,3)(4,7)(5,9)(6,8) ]),
 Group([ (1,4,7)(2,5,8)(3,6,9), (1,2,3)(4,5,6)(7,8,9), (2,5,6,7,3,9,8,4) ]) ]
gap> List(last, IdGroup);
[ [ 72, 40 ], [ 72, 41 ], [ 72, 39 ] ]
</pre>

<a name = "SSEC003.2"></a>
<li><code>OnePrimitiveSolublePermGroup(</code><var>func_1</var><code>, </code><var>arg_1</var><code>, </code><var>func_2</var><code>, </code><var>arg_2</var><code>, ...) F</code>
<a name = "SSEC003.2"></a>
<li><code>OnePrimitiveSolvablePermGroup(</code><var>func_1</var><code>, </code><var>arg_1</var><code>, </code><var>func_2</var><code>, </code><var>arg_2</var><code>, ...) F</code>

This function returns one primitive soluble permutation
group <var>G</var> corresponding to irreducible matrix groups in the IRREDSOL library
for which the return value of <var><var>funci</var>(G)</var> lies in
<var>arg_i</var>, or <code>fail</code> if no such group exists. The arguments <var>func_1</var>, <var>func_2</var>, ...,
must be GAP functions which take a permutation group as their only argument and return a
value, and <var>arg_1</var>, <var>arg_2</var>,
..., must be lists. If <var>arg_i</var> is not a list, <var>arg_i</var> is replaced by the list
<code>[</code><var>arg_i</var><code>]</code>. One of the functions must be <code>Degree</code> or one of its, equivalents, <code>NrMovedPoints</code> or <code>LargestMovedPoint</code>.

For a list of functions which are handled particularly efficiently, see
<a href="CHAP005.htm#SSEC003.1">AllPrimitiveSolublePermGroups</a>.

<pre>
gap> OnePrimitiveSolublePermGroup(Degree, [1..100], Size, [123321]);
fail
</pre>

<a name = "SSEC003.3"></a>
<li><code>IteratorPrimitivePermGroups(</code><var>func_1</var><code>, </code><var>arg_1</var><code>, </code><var>func_2</var><code>, </code><var>arg_2</var><code>, ...) F</code>

This function returns an iterator which runs through the list of all primitive soluble
groups <var>G</var> in the IRREDSOL library such that
<var><var>funci</var>(G)</var> lies in <var>arg_i</var>. The arguments <var>func_1</var>, <var>func_2</var>, ...,
must be GAP functions taking a pc group as their only argument and returning
a value, and <var>arg_1</var>, <var>arg_2</var>, ...,
must be lists. If <var>arg_i</var> is not a list, <var>arg_i</var> is replaced by the list <code>[</code><var>arg_i</var><code>]</code>.
One of the functions must be <code>Degree</code> or one of its, equivalents, <code>NrMovedPoints</code>
or <code>LargestMovedPoint</code>.
For a list of functions which are handled particularly efficiently, see
<a href="CHAP005.htm#SSEC003.1">AllPrimitiveSolublePermGroups</a>.

Using

<code>IteratorPrimitiveSolublePermGroups</code>(<var>func_1</var>, <var>arg_1</var>, <var>func_2</var>, <var>arg_2</var>, ...))

is functionally equivalent to

<code>Iterator</code>(<code>AllPrimitiveSolublePermGroups</code>(<var>func_1</var>, <var>arg_1</var>, <var>func_2</var>, <var>arg_2</var>, ...))

(see <a href="../../../doc/ref/chap30.html#X85A3F00985453F95">Iterators</a> for details) but does not
compute all relevant permutation groups at the same time.


<h2><a name="SECT004">5.4 Recognising primitive soluble groups</a></h2>

<a name = "I8"></a>

<a name = "I9"></a>

<a name = "I10"></a>

<a name = "I11"></a>

<a name = "I12"></a>

<a name = "I13"></a>

<a name = "I14"></a>

<a name = "I15"></a>

<a name = "SSEC004.1"></a>
<li><code>IdPrimitiveSolubleGroup(</code><var>G</var><code>) F</code>
<a name = "SSEC004.1"></a>
<li><code>IdPrimitiveSolubleGroupNC(</code><var>G</var><code>) F</code>
<a name = "SSEC004.1"></a>
<li><code>IdPrimitiveSolvableGroup(</code><var>G</var><code>) F</code>
<a name = "SSEC004.1"></a>
<li><code>IdPrimitiveSolvableGroupNC(</code><var>G</var><code>) F</code>

returns the id of the primitive soluble group <var>G</var>. This is the same as
the id of <code>IrreducibleMatrixGroupPrimitiveSolubleGroup</code>(<var>G</var>), see <a href="CHAP005.htm#SSEC001.4">IrreducibleMatrixGroupPrimitiveSolubleGroup</a> and <a href="CHAP003.htm#SSEC001.3">IdIrreducibleSolubleMatrixGroup</a>.
Note that two primitive soluble groups are isomorphic if, and only if, their
ids returned by <code>IdPrimitivePcGroup</code> are the same. The NC version does not
check whether <var>G</var> is primitive and soluble.

<pre>
gap> G := SmallGroup(432, 734);
<pc group of size 432 with 7 generators>
gap> IdPrimitiveSolubleGroup(G);
[ 2, 3, 1, 2 ]
gap> G := AlternatingGroup(4);
Alt( [ 1 .. 4 ] )
gap> IdPrimitiveSolubleGroup(G);
[ 2, 2, 2, 1 ]
</pre>

<a name = "SSEC004.2"></a>
<li><code>RecognitionPrimitiveSolubleGroup(</code><var>G</var><code>,</code><var>wantiso</var><code>) F</code>
<a name = "SSEC004.2"></a>
<li><code>RecognitionPrimitiveSolvableGroup(</code><var>G</var><code>,</code><var>wantiso</var><code>) F</code>

This function returns a record <var>r</var> which identifies the primitive soluble group <var>G</var>.
The component <code>id</code> is always present and contains the id of <var>G</var>. if <var>wantiso</var> is <code>true</code>, then
the component <code>iso</code> is bound to an isomorphism from <var>G</var> into a primitive pc group.


<h2><a name="SECT005">5.5 Obsolete functions</a></h2>

<a name = "I16"></a>

<a name = "SSEC005.1"></a>
<li><code>PrimitiveSolublePermutationGroup(</code><var>n</var><code>,</code><var>q</var><code>,</code><var>d</var><code>,</code><var>k</var><code>) F</code>
<a name = "SSEC005.1"></a>
<li><code>PrimitiveSolvablePermutationGroup(</code><var>n</var><code>,</code><var>q</var><code>,</code><var>d</var><code>,</code><var>k</var><code>) F</code>
<a name = "SSEC005.1"></a>
<li><code>PrimitivePermutationGroupIrreducibleMatrixGroup(</code><var>G</var><code>) F</code>
<a name = "SSEC005.1"></a>
<li><code>PrimitivePermutationGroupIrreducibleMatrixGroupNC(</code><var>G</var><code>) F</code>
<a name = "SSEC005.1"></a>
<li><code>AllPrimitiveSolublePermutationGroups(</code><var>func_1</var><code>, </code><var>arg_1</var><code>, </code><var>func_2</var><code>, </code><var>arg_2</var><code>, ...) F</code>
<a name = "SSEC005.1"></a>
<li><code>AllPrimitiveSolvablePermutationGroups(</code><var>func_1</var><code>, </code><var>arg_1</var><code>, </code><var>func_2</var><code>, </code><var>arg_2</var><code>, ...) F</code>
<a name = "SSEC005.1"></a>
<li><code>IteratorPrimitivePermutationGroups(</code><var>func_1</var><code>, </code><var>arg_1</var><code>, </code><var>func_2</var><code>, </code><var>arg_2</var><code>, ...) F</code>
<a name = "SSEC005.1"></a>
<li><code>OnePrimitiveSolublePermutationGroup(</code><var>func_1</var><code>, </code><var>arg_1</var><code>, </code><var>func_2</var><code>, </code><var>arg_2</var><code>, ...) F</code>
<a name = "SSEC005.1"></a>
<li><code>OnePrimitiveSolvablePermutationGroup(</code><var>func_1</var><code>, </code><var>arg_1</var><code>, </code><var>func_2</var><code>, </code><var>arg_2</var><code>, ...) F</code>

These functions have been renamed from ...<code>PermutationGroup</code>...to ...<code>PermGroup</code>....
The above function names are deprecated.


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