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I guess my question is rather soft. I'm looking for "interesting" infinite sequences of finite groups $$ \Gamma=(G_i, G_{i+1}, \cdots, G_n, \cdots) $$ such that $|G_n|=nk+r$ for some constants $k,r$. Here $i$ is just some initial index. Of course one can just put completely unrelated groups in the above sequence of the given order. I'm looking for, however for "nice" families of groups with the above property. Examples are

  • The cyclic sequence $(\mathbb{Z}_2, \mathbb{Z}_3, \cdots)$, here $p=2$, $k=1$ and $r=0$.
  • Dihedral sequence $(\mathrm{Dih}_2, \mathrm{Dih}_3, \cdots)$, here $p=2$ and $k=2$, $r=0$.
  • Dicyclic sequence $(\mathrm{Di}_2, \mathrm{Di}_3, \cdots)$, here again $p=2$ and $r=0$ but $k=4$ .
  • Of course if $\Gamma$ is a "nice" family and $H$ is any group then $H\times \Gamma$ will also work.

Are they more interesting examples? Sorry I cannot make precise what I mean by words like "interesting" and "nice", making the question a bit (OK actually very) subjective.

Motivation: I'm not sure how much this will help. Originally the question comes from graph theory. I'm looking from infinite sequences of Cayley graphs of the form $$ C=(\mathrm{Cay}(G_i,S_i), \mathrm{Cay}(G_{i+1},S_{i+1}), \cdots, \mathrm{Cay}(G_n, S_n), \cdots) $$ here $S_n\subset G_n$ is an inverse-closed subset. This sequence must satisfy the following properties:

  • $|G_n|=|\mathrm{Cay}(G_n, S_n)|=nk+r$ for some constants $k,r$.
  • The degree of $\mathrm{Cay}(G_n, S_n)$ also grows linearly; i.e. $|S_n|=ns+t$ again with $s,t$ constant.
  • One has (many) inclusion(s) $\mathrm{Cay}(G_n,S_n)\subset \mathrm{Cay}(G_{n+1},S_{n+1})$.

There are even more conditions like $\mathrm{Cay}(G_n, S_n)$ is connected ($S_n$ is a generating set); chromatic number of $\mathrm{Cay}(G_n, S_n)$ must be exactly $n$ and furthermore $\mathrm{Cay}(G_n, S_n)$ must be uniquely $n$-colorable. I have constructed such examples for the cyclic and dihedral sequence. Pretty sure there is no example (that I care about) for dicyclic too. Looking for some food for thought about what other possibilities I can explore.

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    "A bit subjective" seems to be an understatement, but you can make it better if you tell us what's this question's background and/or what do you want this for. – 2017-02-22
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    @DonAntonio I have added the motivation. You're right it is very subjective. – 2017-02-22
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    You can generalise your fourth point - if $(G_i,G_{i+1},\ldots)$ and $(H_i,H_{i+1},\ldots)$ are any two such sequences then $(G_i\rtimes H_i,\ldots)$ is such a sequence, where $G_i\rtimes H_i$ can be any semidirect product. Dihedral and Dicyclic groups can be obtained in this way from the cyclic groups. – 2017-02-22

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