# Imprimitive group

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 $V$).

How to Cite This Entry:
Imprimitive group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Imprimitive_group&oldid=52130
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