  We prove that a generic $k$-parameter bifurcation of a dynamical system with
a monoid symmetry occurs along a generalized kernel or center subspace of a
particular type. More precisely, any (complementable) subrepresentation $U$ is
given a number $K_U$ and a number $C_U$. A $k$-parameter bifurcation can
generically only occur along a generalized kernel isomorphic to $U$ if $k \geq
K_U$. It can generically only occur along a center subspace isomorphic to $U$
if $k \geq C_U$. The numbers $K_U$ and $C_U$ depend only on the decomposition
of $U$ into indecomposable subrepresentations. In particular, we prove that a
generic one-parameter steady-state bifurcation occurs along one absolutely
indecomposable subrepresentation. Likewise, it follows that a generic
one-parameter Hopf bifurcation occurs along one indecomposable
subrepresentation of complex or quaternionic type, or along two isomorphic
absolutely indecomposable subrepresentations. In order to prove these results,
we show that the set of endomorphisms with generalized kernel (or center
subspace) isomorphic to $U$ is the disjoint union of a finite set of conjugacy
invariant submanifolds of codimension $K_U$ and higher (or $C_U$ and higher).
The results in this article hold for any monoid, including non-compact groups.

Dynamical Systems