The main object of this work is to find an explicit formula
for the action of a finite group on a commutative ring. Let *G* be a
finite group which acts upon a commutative ring *k*. We can define the
trace map *tr*_{G}:k *à
k*^{G}, and ask ourselves when this map is onto. This question is of
interest for us, because it turns out that *tr*_{G} is onto if and
only if *k* is projective as a module over the twisted group ring *k*_{t}G.
It turns out, since *tr*_{G}_{ }is_{ }*k*^{G}
linear, that *tr*_{G}_{ }is onto if and only if there
exists an element *x*_{G}_{ }in *k*, such that *tr*_{G}(x_{G})=1.
It is known that if *tr*_{G}_{ }is onto, then *tr*_{H}_{
}is onto, for every subgroup *H* of *G*. The main focus of this
work will be to deduce the other direction: Suppose we know that for some of
the subgroups *H* of *G*, *tr*_{H}_{ }is onto.
When can we deduce that *tr*_{G} is onto as well? Equivalently, we
may ask when the existences of *x*_{H}, for some of the subgroups *H*
of *G* determine the existence of *x*_{G}. It is known, for example,
that if *H* is a subgroup of *G* such that there exists an element *x*_{H},
and for every subgroup *N* of *G* that intersects *H* trivially,
there exists an element *x*_{N}, then there exists an element *x*_{G}.
Moreover, there exists a polynomial formula which expresses *x*_{G}
in terms of the elements *x*_{H} and *x*_{N}. In this
work we will find a formula for *x*_{G} in terms of the elements *x*_{P},
where *P* varies over the subgroup of *G* of prime order. We will
also give several examples for using the two formulas mentioned above. In
addition, we will discuss the case where the ring *k* is not commutative.
It is known that in this case, the existence of *x*_{E}, for every
elementary abelian subgroup *E* of *G*, determines the existence of *x*_{G}.
However, there is not a known formula for *x*_{G} in terms of the *x*_{E}'s.
We shall give such a formula in the case where *G* is the cyclic group of
order *p*^{2}, for *p* a prime number. We will also show that
the existence (or non-existence) of some of the *x*_{P}'s, does
not determine the existence (or non-existence) of the others.