## Special symmetry classes

We now return to *Y*^{X} and consider its subsets consisting of the injective
and the surjective maps *f* only:

*Y*^{X}_{i}:= {f ÎY^{X} | f injective } and
Y_{s}^{X}:= {f ÎY^{X} | f surjective }.

It is clear
that each of these sets is both a *G*-set and an *H*-set and therefore it
is also an *H ´G*-set, but it will not in general
be an * H wr *_{X} G -set.
The corresponding orbits of *G,H* and *H ´G* on *Y*_{i}^{X} are called
*injective*
symmetry classes, while those on *Y*_{s}^{X} will be called
*surjective*
symmetry classes. We should like to determine their number.
In order to do this we describe the fixed points
of *(h,g) ÎH ´G* on these sets to prepare an application of the
Cauchy-Frobenius Lemma. A first remark shows how the fixed points of *(h,g)*
on *Y*^{X} can be constructed with the aid of * bar (h)* and * bar (g)*,
the permutations induced by *h* on *Y* and by *g* on *X* (use lemma):

**Corollary: **
*
If ** bar (g)= Õ*_{ n}(x_{ n} ...g^{l n-1}x_{ n})
, then *f ÎY*^{X} is fixed under *(h,g)* if and only if
the following two conditions
are satisfied:
*f(x*_{ n}) ÎY_{hl n},

and
the other values of *f* arise from
the values *f(x*_{ n}) according to
*f(x*_{ n})=hf(g^{-1}x_{ n})=h^{2}f(
g^{-2}x_{ n})= ... .

This together with lemma yields:

**Corollary: **
*
The fixed points of **(h,g)* are
the *f ÎY*^{X} which can be
obtained in the following way:
- To each cyclic factor of
* bar (g)*, let *l* denote its length, we
associate a cyclic factor of
* bar (h)* of length *d* dividing *l*.
- If
*x* is a point in this cyclic factor
of * bar (g)* and *y* a point
in the chosen cyclic factor of * bar (h)*, then put
*f(x):=y, f(gx):=hy,f(g*^{2}x):=h^{2}y,
... .

last changed: January 19, 2005