  Consider any sequence of finite groups $A^t$, where $t$ takes values in an
integer index set $\mathbf{Z}$. A group system $A$ is a set of sequences with
components in $A^t$ that forms a group under componentwise addition in $A^t$,
for each $t\in\mathbf{Z}$. In the setting of group systems, a natural
definition of a linear system is a homomorphism from a group of inputs to an
output group system $A$. We show that any group can be the input group of a
linear system and some group system. In general the kernel of the homomorphism
is nontrivial. We show that any $\ell$-controllable complete group system $A$
is a linear system whose input group is a generator group
$({\mathcal{U}},\circ)$, deduced from $A$, and then the kernel is always
trivial. The input set ${\mathcal{U}}$ is a set of tensors, a double Cartesian
product space of sets $R_{0,k}^t$, with indices $k$, for $0\le k\le\ell$, and
time $t$, for $t\in\mathbf{Z}$. $R_{0,k}^t$ is a set of unique generator labels
for the generators in $A$ with nontrivial span for the time interval $[t,t+k]$.
We show the generator group contains an elementary system, an infinite
collection of elementary groups, one for each $k$ and $t$, defined on small
subsets of ${\mathcal{U}}$, in the shape of triangles, which form a tile like
structure over ${\mathcal{U}}$. There is a homomorphism from each elementary
group to any elementary group defined on smaller tiles of the former group. Any
elementary system is sufficient to define a unique generator group up to
isomorphism, and therefore is sufficient to construct a linear system and group
system as well. Any linear block code is a strongly controllable group system.
Then we can obtain new results on the structure of block codes using the
generator group. There is a harmonic theory of group systems which we study
using the generator group.
