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A group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i0503501.png" /> of one-to-one mappings (permutations, cf. [[Permutation|Permutation]]) of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i0503502.png" /> onto itself, for which there exists a partition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i0503503.png" /> into a union of disjoint subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i0503504.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i0503505.png" />, with the following properties: the number of elements in at least one of the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i0503506.png" /> is greater than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i0503507.png" />; for any permutation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i0503508.png" /> and any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i0503509.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035010.png" />, there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035012.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035013.png" /> maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035014.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035015.png" />.
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The collection of subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035016.png" /> is called a system of imprimitivity, while the subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035017.png" /> themselves are called domains of imprimitivity of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035018.png" />. A non-imprimitive group of permutations is called primitive.
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An example of an imprimitive group is a non-trivial intransitive group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035019.png" /> of permutations of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035020.png" /> (see [[Transitive group|Transitive group]]): for a system of imprimitivity one can take the collection of all orbits (domains of transitivity, cf. [[Orbit|Orbit]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035021.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035022.png" />. A transitive group of permutations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035023.png" /> of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035024.png" /> is primitive if and only if for some element (and hence for all elements) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035025.png" /> the set of permutations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035026.png" /> leaving <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035027.png" /> fixed is a maximal subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035028.png" />.
+
A group  $  G $
 +
of one-to-one mappings (permutations, cf. [[Permutation|Permutation]]) of a set  $  S $
 +
onto itself, for which there exists a partition of $  S $
 +
into a union of disjoint subsets  $  S _ {1} \dots S _ {m} $,
 +
$  m \geq  2 $,  
 +
with the following properties: the number of elements in at least one of the sets  $  S _ {i} $
 +
is greater than  $  1 $;
 +
for any permutation  $  g \in G $
 +
and any  $  i $,
 +
$  1 \leq  i \leq  m $,
 +
there exists a $  j $,
 +
$  1 \leq  j \leq  m $,
 +
such that  $  g $
 +
maps  $  S _ {i} $
 +
onto  $  S _ {j} $.
  
The notion of an imprimitive group of permutations has an analogue for groups of linear transformations of vector spaces. Namely, a [[Linear representation|linear representation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035029.png" /> of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035030.png" /> is called imprimitive if there exists a decomposition of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035031.png" /> of the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035032.png" /> into a direct sum of proper subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035033.png" /> with the following property: For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035034.png" /> and any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035036.png" />, there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035038.png" />, such that
+
The collection of subsets  $  S _ {1} \dots S _ {m} $
 +
is called a system of imprimitivity, while the subsets  $  S _ {i} $
 +
themselves are called domains of imprimitivity of the group  $  G $.  
 +
A non-imprimitive group of permutations is called primitive.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035039.png" /></td> </tr></table>
+
An example of an imprimitive group is a non-trivial intransitive group  $  G $
 +
of permutations of a set  $  S $(
 +
see [[Transitive group|Transitive group]]): for a system of imprimitivity one can take the collection of all orbits (domains of transitivity, cf. [[Orbit|Orbit]]) of  $  G $
 +
on  $  S $.  
 +
A transitive group of permutations  $  G $
 +
of a set  $  S $
 +
is primitive if and only if for some element (and hence for all elements)  $  y \in S $
 +
the set of permutations of  $  G $
 +
leaving  $  y $
 +
fixed is a maximal subgroup of  $  G $.
  
The collection of subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035040.png" /> is called a system of imprimitivity of the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035041.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035042.png" /> does not have a decomposition of the above type, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035043.png" /> is said to be a primitive representation. An imprimitive representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035044.png" /> is called transitive imprimitive if there exists for any pair of subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035045.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035046.png" /> of the system of imprimitivity an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035047.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035048.png" />. The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035049.png" /> of linear transformations of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035050.png" /> and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035051.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035052.png" /> defined by the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035053.png" /> are also called imprimitive (or primitive) if the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035054.png" /> is imprimitive (or primitive).
+
The notion of an imprimitive group of permutations has an analogue for groups of linear transformations of vector spaces. Namely, a [[Linear representation|linear representation]]  $  \rho $
 +
of a group  $  G $
 +
is called imprimitive if there exists a decomposition of the space  $  V $
 +
of the representation  $  \rho $
 +
into a direct sum of proper subspaces  $  V _ {1} \dots V _ {m} $
 +
with the following property: For any  $  g \in G $
 +
and any  $  i $,
 +
$  1 \leq  i \leq  m $,
 +
there exists a  $  j $,
 +
$  1 \leq  j \leq  m $,
 +
such that
  
Examples. A representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035055.png" /> of the symmetric group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035056.png" /> in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035057.png" />-dimensional vector space over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035058.png" /> that rearranges the elements of a basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035059.png" /> is transitive imprimitive, the one-dimensional subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035060.png" /> form a system of imprimitivity for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035061.png" />. Another example of a transitive imprimitive representation is the [[Regular representation|regular representation]] of a finite group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035062.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035063.png" />; the collection of one-dimensional subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035064.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035065.png" /> runs through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035066.png" />, forms a system of imprimitivity. More generally, any [[Monomial representation|monomial representation]] of a finite group is imprimitive. The representation of a cyclic group of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035067.png" /> by rotations of the real plane through angles that are multiples of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035068.png" /> is primitive.
+
$$
 +
\rho ( g) ( V _ {i} )  = \
 +
V _ {j} .
 +
$$
  
The notion of an imprimitive representation is closely related to that of an [[Induced representation|induced representation]]. Namely, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035069.png" /> be an imprimitive finite-dimensional representation of a finite group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035070.png" /> with system of imprimitivity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035071.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035072.png" /> is partitioned into a union of orbits with respect to the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035073.png" /> determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035074.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035075.png" /> be a complete set of representatives of the different orbits of this action, let
+
The collection of subsets  $  V _ {1} \dots V _ {m} $
 +
is called a system of imprimitivity of the representation $  \rho $.  
 +
If  $  V $
 +
does not have a decomposition of the above type, then  $  \rho $
 +
is said to be a primitive representation. An imprimitive representation $  \rho $
 +
is called transitive imprimitive if there exists for any pair of subspaces  $  V _ {i} $
 +
and  $  V _ {j} $
 +
of the system of imprimitivity an element  $  g \in G $
 +
such that  $  \rho ( g) ( V _ {i} ) = V _ {j} $.  
 +
The group  $  \rho ( G) $
 +
of linear transformations of the space  $  V $
 +
and the  $  G $-
 +
module  $  V $
 +
defined by the representation  $  \rho $
 +
are also called imprimitive (or primitive) if the representation  $  \rho $
 +
is imprimitive (or primitive).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035076.png" /></td> </tr></table>
+
Examples. A representation  $  \rho $
 +
of the symmetric group  $  S _ {n} $
 +
in the  $  n $-
 +
dimensional vector space over a field  $  k $
 +
that rearranges the elements of a basis  $  e _ {1} \dots e _ {n} $
 +
is transitive imprimitive, the one-dimensional subspaces  $  \{ k e _ {1} \dots k e _ {n} \} $
 +
form a system of imprimitivity for  $  \rho $.
 +
Another example of a transitive imprimitive representation is the [[Regular representation|regular representation]] of a finite group  $  G $
 +
over a field  $  k $;  
 +
the collection of one-dimensional subspaces  $  k g $,
 +
where  $  g $
 +
runs through  $  G $,
 +
forms a system of imprimitivity. More generally, any [[Monomial representation|monomial representation]] of a finite group is imprimitive. The representation of a cyclic group of order  $  m \geq  3 $
 +
by rotations of the real plane through angles that are multiples of  $  2 \pi / m $
 +
is primitive.
  
let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035077.png" /> be the representation of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035078.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035079.png" /> defined by the restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035080.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035081.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035082.png" /> be the representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035083.png" /> induced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035084.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035085.png" /> is equivalent to the direct sum of the representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035086.png" />. Conversely, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035087.png" /> be any collection of subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035088.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035089.png" /> be a representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035090.png" /> in a finite-dimensional vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035091.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035092.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035093.png" /> be the representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035094.png" /> induced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035095.png" />. Suppose further that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035096.png" /> is a system of representatives of left cosets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035097.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035098.png" />. Then the direct sum of the representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i05035099.png" /> is imprimitive, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i050350100.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i050350101.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i050350102.png" />, is a system of imprimitivity (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i050350103.png" /> is canonically identified with a subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050350/i050350104.png" />).
+
The notion of an imprimitive representation is closely related to that of an [[Induced representation|induced representation]]. Namely, let $  \rho $
 +
be an imprimitive finite-dimensional representation of a finite group  $  G $
 +
with system of imprimitivity  $  \{ V _ {1} \dots V _ {n} \} $.  
 +
The set  $  \{ V _ {1} \dots V _ {n} \} $
 +
is partitioned into a union of orbits with respect to the action of  $  G $
 +
determined by  $  \rho $.  
 +
Let  $  \{ V _ {i _ {1}  } \dots V _ {i _ {s}  } \} $
 +
be a complete set of representatives of the different orbits of this action, let
 +
 
 +
$$
 +
H _ {m}  = \
 +
\{ {g \in G } : {\rho ( g) ( V _ {i _ {m}  } ) = V _ {i _ {m}  } } \}
 +
,\  m = 1 \dots s ,
 +
$$
 +
 
 +
let  $  \phi _ {m} $
 +
be the representation of the group  $  H _ {m} $
 +
in  $  V _ {i _ {m}  } $
 +
defined by the restriction of $  \rho $
 +
to $  H _ {m} $,  
 +
and let $  \rho _ {m} $
 +
be the representation of $  G $
 +
induced by $  \phi _ {m} $.  
 +
Then $  \rho $
 +
is equivalent to the direct sum of the representations $  \rho _ {1} \dots \rho _ {s} $.  
 +
Conversely, let $  H _ {1} \dots H _ {s} $
 +
be any collection of subgroups of $  G $,  
 +
let $  \phi _ {m} $
 +
be a representation of $  H _ {m} $
 +
in a finite-dimensional vector space $  W _ {m} $,  
 +
$  m = 1 \dots s $,  
 +
and let $  \rho _ {m} $
 +
be the representation of $  G $
 +
induced by $  \phi _ {m} $.  
 +
Suppose further that $  \{ g _ {m,j} \} _ {j=} 1 ^ {r _ {m} } $
 +
is a system of representatives of left cosets of $  G $
 +
with respect to $  H _ {m} $.  
 +
Then the direct sum of the representations $  \rho _ {1} \dots \rho _ {s} $
 +
is imprimitive, while $  \rho ( g _ {m,j} ) ( W _ {m} ) $,  
 +
$  j = 1 \dots r _ {m} $,  
 +
$  m = 1 \dots s $,  
 +
is a system of imprimitivity (here $  W _ {m} $
 +
is canonically identified with a subspace of $  V $).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Hall,  "Group theory" , Macmillan  (1959)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C.W. Curtis,  I. Reiner,  "Representation theory of finite groups and associative algebras" , Interscience  (1962)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Hall,  "Group theory" , Macmillan  (1959)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C.W. Curtis,  I. Reiner,  "Representation theory of finite groups and associative algebras" , Interscience  (1962)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
A domain of imprimitivity is also called a block.
 
A domain of imprimitivity is also called a block.

Revision as of 22:12, 5 June 2020


A group $ G $ of one-to-one mappings (permutations, cf. Permutation) of a set $ S $ onto itself, for which there exists a partition of $ S $ into a union of disjoint subsets $ S _ {1} \dots S _ {m} $, $ m \geq 2 $, with the following properties: the number of elements in at least one of the sets $ S _ {i} $ is greater than $ 1 $; for any permutation $ g \in G $ and any $ i $, $ 1 \leq i \leq m $, there exists a $ j $, $ 1 \leq j \leq m $, such that $ g $ maps $ S _ {i} $ onto $ S _ {j} $.

The collection of subsets $ S _ {1} \dots S _ {m} $ is called a system of imprimitivity, while the subsets $ S _ {i} $ themselves are called domains of imprimitivity of the group $ G $. A non-imprimitive group of permutations is called primitive.

An example of an imprimitive group is a non-trivial intransitive group $ G $ of permutations of a set $ S $( see Transitive group): for a system of imprimitivity one can take the collection of all orbits (domains of transitivity, cf. Orbit) of $ G $ on $ S $. A transitive group of permutations $ G $ of a set $ S $ is primitive if and only if for some element (and hence for all elements) $ y \in S $ the set of permutations of $ G $ leaving $ y $ fixed is a maximal subgroup of $ G $.

The notion of an imprimitive group of permutations has an analogue for groups of linear transformations of vector spaces. Namely, a linear representation $ \rho $ of a group $ G $ is called imprimitive if there exists a decomposition of the space $ V $ of the representation $ \rho $ into a direct sum of proper subspaces $ V _ {1} \dots V _ {m} $ with the following property: For any $ g \in G $ and any $ i $, $ 1 \leq i \leq m $, there exists a $ j $, $ 1 \leq j \leq m $, such that

$$ \rho ( g) ( V _ {i} ) = \ V _ {j} . $$

The collection of subsets $ V _ {1} \dots V _ {m} $ is called a system of imprimitivity of the representation $ \rho $. If $ V $ does not have a decomposition of the above type, then $ \rho $ is said to be a primitive representation. An imprimitive representation $ \rho $ is called transitive imprimitive if there exists for any pair of subspaces $ V _ {i} $ and $ V _ {j} $ of the system of imprimitivity an element $ g \in G $ such that $ \rho ( g) ( V _ {i} ) = V _ {j} $. The group $ \rho ( G) $ of linear transformations of the space $ V $ and the $ G $- module $ V $ defined by the representation $ \rho $ are also called imprimitive (or primitive) if the representation $ \rho $ is imprimitive (or primitive).

Examples. A representation $ \rho $ of the symmetric group $ S _ {n} $ in the $ n $- dimensional vector space over a field $ k $ that rearranges the elements of a basis $ e _ {1} \dots e _ {n} $ is transitive imprimitive, the one-dimensional subspaces $ \{ k e _ {1} \dots k e _ {n} \} $ form a system of imprimitivity for $ \rho $. Another example of a transitive imprimitive representation is the regular representation of a finite group $ G $ over a field $ k $; the collection of one-dimensional subspaces $ k g $, where $ g $ runs through $ G $, forms a system of imprimitivity. More generally, any monomial representation of a finite group is imprimitive. The representation of a cyclic group of order $ m \geq 3 $ by rotations of the real plane through angles that are multiples of $ 2 \pi / m $ is primitive.

The notion of an imprimitive representation is closely related to that of an induced representation. Namely, let $ \rho $ be an imprimitive finite-dimensional representation of a finite group $ G $ with system of imprimitivity $ \{ V _ {1} \dots V _ {n} \} $. The set $ \{ V _ {1} \dots V _ {n} \} $ is partitioned into a union of orbits with respect to the action of $ G $ determined by $ \rho $. Let $ \{ V _ {i _ {1} } \dots V _ {i _ {s} } \} $ be a complete set of representatives of the different orbits of this action, let

$$ H _ {m} = \ \{ {g \in G } : {\rho ( g) ( V _ {i _ {m} } ) = V _ {i _ {m} } } \} ,\ m = 1 \dots s , $$

let $ \phi _ {m} $ be the representation of the group $ H _ {m} $ in $ V _ {i _ {m} } $ defined by the restriction of $ \rho $ to $ H _ {m} $, and let $ \rho _ {m} $ be the representation of $ G $ induced by $ \phi _ {m} $. Then $ \rho $ is equivalent to the direct sum of the representations $ \rho _ {1} \dots \rho _ {s} $. Conversely, let $ H _ {1} \dots H _ {s} $ be any collection of subgroups of $ G $, let $ \phi _ {m} $ be a representation of $ H _ {m} $ in a finite-dimensional vector space $ W _ {m} $, $ m = 1 \dots s $, and let $ \rho _ {m} $ be the representation of $ G $ induced by $ \phi _ {m} $. Suppose further that $ \{ g _ {m,j} \} _ {j=} 1 ^ {r _ {m} } $ is a system of representatives of left cosets of $ G $ with respect to $ H _ {m} $. Then the direct sum of the representations $ \rho _ {1} \dots \rho _ {s} $ is imprimitive, while $ \rho ( g _ {m,j} ) ( W _ {m} ) $, $ j = 1 \dots r _ {m} $, $ m = 1 \dots s $, is a system of imprimitivity (here $ W _ {m} $ is canonically identified with a subspace of $ V $).

References

[1] M. Hall, "Group theory" , Macmillan (1959)
[2] C.W. Curtis, I. Reiner, "Representation theory of finite groups and associative algebras" , Interscience (1962)

Comments

A domain of imprimitivity is also called a block.

How to Cite This Entry:
Imprimitive group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Imprimitive_group&oldid=47323
This article was adapted from an original article by N.N. Vil'yamsV.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article