What I mean by "primeless isomorphism" is essentially a relation on finite groups by identifying groups whose structure differs only in which primes divide the groups' orders. The groups aren't exactly isomorphic, but they are close to it; I'm having trouble formulating the idea rigorously, so I think it might be best to explain what I mean through examples.
In the simplest case, take cyclic groups $C_{p}$ and $C_q$ for distinct primes $p$ and $q$. Obviously these aren't isomorphic, but they would be "primeless isomorphic" as the only difference in the group structure is what the primes actually are. By contrast, $C_{p}$ and $C_n$ would not be considered primeless isomorphic for composite $n$, because $C_n$ has proper nontrivial subgroups and $C_p$ doesn't - a structural difference independent of which primes divide $n$.
For another example, we could look at the class of Frobenius groups $C_pC_q$ where $C_q$ acts fixed point freely on $C_p$ (again with distinct primes $p,q$). There are constraints on what these primes can be in that $q$ has to divide $p-1$ for the group to exist, but among those groups that do, it doesn't seem like they are qualitatively different. Groups of order $p^3$ have been classified in exactly the way that I mean; the same thing goes for Dihedral groups $D_{2n}$ of squarefree order, which split into different "primeless isomorphism classes" depending on the number of prime divisors.
The set of all groups with order $p^3$ for some prime $p$ would be divided into seven equivalency classes: $[C_{p^3}],[C_{p^2}\times C_p],[(C_p)^3],[Q_8],[D_8],[\text{Heis}\,Z_p],$ and $[G_p]$ (where for the last two classes $p$ is odd).
It's hard to say precisely what I mean, but hopefully you get my drift.
Has it been studied? Is there a name for it?
If not, is that because it is somehow logically difficult (or impossible) to define?
If nobody's heard of this,
- Can anyone think of a good way to formulate a definition for two groups to be "primeless isomorphic?" The definition should give rise to an equivalence relation on any given set of groups that partitions it into classes which are only "quantitatively" different, but not "qualitatively" in the way I've been getting at.