The following proposition is a tool for constructing permutation representations out of a given permutation representation f: G -> Sym(X).
Proposition
- If H is a subgroup of G, then the restriction of f to H is also a permutation representation.
-
Let Y be a subset of X such that for all g
G and all y Y,
also g(y)
Y. Then every g G determines by restriction to Y
a bijection g' of Y. The resulting map
G -> Sym(Y), g -> g'
is a permutation representation.
A set Y as in the proposition is called invariant under G. We also say Y is G-invariant or, if G is clear from the context, just invariant.