A number of years ago I studied a preon model (Journal of Mathematical Physics 38:3414-3426, 1997) in which the preons interacted like group elements. I thought it curious that you could sometimes define continuous transformations over the group elements which preserved the group structure.
Specifically, find a continuous family of V such that [edit] if we define $g_i$ as the elements of the finite group,
${g_i}' = V g_i$
if $g_i g_j = g_k$ then ${g_i}' {g_j}' = {g_k}'$
[edit] V is continuous in the sense that the difference between two different transformations can be arbitrarily small. If represented as a matrix, the matrix elements can be arbitrarily small. I know there can be permutations, for which this isn't true, but I wasn't interested in those. I was representing the group elements as unit vectors in N-space where N is the group size, and the transformations as NxN complex matrices.
I was using complex numbers, and in that case at least, there aren't any such continuous transformations if the finite group is abelian. (That's why I don't think the model is physical.) Some non-abelian groups admit transformations isomorphic to SU(2) and SU(3), and of other things.
I'm confident that this is well-known, but I'm not deeply familiar with the research in group algebras, and I'm not sure where to look to find out more--e.g. if there's a pattern to this.
[edit] My apologies for a lack of precision in the initial question.
[edit2] Maybe a concrete example is in order The simplest non-abelian group has the Cayley table that I specify as $\begin{array}{c|cccccc} X&0&1&2&3&4&5 \\ \hline 0&0&1&2&3&4&5 \\ 1&1&2&0&4&5&3 \\ 2&2&0&1&5&3&4 \\ 3&3&5&4&0&2&1 \\ 4&4&3&5&1&0&2 \\ 5&5&4&3&2&1&0 \end{array}$
If an element in the group algebra is represented by a vector, with $g_0$ = (1,0,0,0,0,0), $g_1$=(0,1,0,0,0,0), etc, then if a, b, and c are infinitesimal small changes, $\bar{1}$ + $\delta V$ is a transformation which, when applied to the group elements, gives new elements whose products are isomorphic to those of the original group elements. $\bar{1}$ is the identity 6x6 matrix, and
$\delta V$ = $ \begin{array}{cccccc} 0&0&0 &0&0&0\\ 0&0&0 &a&-b&-a+b\\ 0&0&0 &-a&b&a-b\\ 0&a&-a& 0&c&-c \\ 0&-b&b&-c&0&c \\ 0&-a+b&a-b&c&-c&0 \\ \end{array}$
