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Consider any sequence of finite groups $A^t$, where $t$ takes values in aninteger index set $\mathbf{Z}$. A group system $A$ is a set of sequences withcomponents 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 naturaldefinition of a linear system is a homomorphism from a group of inputs to anoutput group system $A$. We show that any group can be the input group of alinear system and some group system. In general the kernel of the homomorphismis 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 alwaystrivial. The input set ${\mathcal{U}}$ is a set of tensors, a double Cartesianproduct space of sets $R_{0,k}^t$, with indices $k$, for $0\le k\le\ell$, andtime $t$, for $t\in\mathbf{Z}$. $R_{0,k}^t$ is a set of unique generator labelsfor the generators in $A$ with nontrivial span for the time interval $[t,t+k]$.We show the generator group contains an elementary system, an infinitecollection of elementary groups, one for each $k$ and $t$, defined on smallsubsets of ${\mathcal{U}}$, in the shape of triangles, which form a tile likestructure over ${\mathcal{U}}$. There is a homomorphism from each elementarygroup to any elementary group defined on smaller tiles of the former group. Anyelementary system is sufficient to define a unique generator group up toisomorphism, and therefore is sufficient to construct a linear system and groupsystem 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 thegenerator group. There is a harmonic theory of group systems which we studyusing the generator group.

Any strongly controllable group system or group shift or any linear block code is a linear system whose input is a generator group