Differences between $p$-groups and $q$-groups

Question

First, let me include the same disclaimer that goes in the first line of any article I write: all groups considered herein are finite.

Academically, I work with connecting the arithmetic structure of the set of character degrees $\bigl\{\ \chi(1)\ \big|\ \chi \in \operatorname{Irr}_{\mathbb C}(G)\ \bigr\}$ with the underlying group structure of a group $G$. In this setting, nonabelian $p$-groups are (generally speaking) a triviality which are quickly dispensed in favor of getting to more interesting considerations.

Which is to say, my general understanding of nilpotent groups is not where I want it to be. I certainly know the foundational, easier results, most of which are true across all primes (except possibly $p=2$ and occasionally $p=3$). The general theme of this discussion is meant to be: what is different/peculiar about the prime $p$ that makes $p$-groups different from $q$-groups over primes $q\ne p$, and when/in what ways/etc. does this difference manifest? With that, the specific question is:

What are ways that $p$-groups and $q$-groups behave differently for distinct primes $p$ and $q$?

This is very much intended to be a Big List question and will be requested into a community wiki accordingly.

As examples of what is being requested, consider the following context and subsequent questions. Independent of the specific value of $p$ and up to isomorphism,

• there is exactly one group of order $p$,

• there are exactly two groups of order $p^2$,

• there are exactly five groups of order $p^3$ (three abelian and two extraspecial).

1. If it exists, what is a value $k$ for which there are different numbers of (isomorphism classes of) groups of order $p^k$ and $q^k$?

2. Is the smallest such $k$ known?

3. If the smallest $k$ is known, are the smallest primes $p$, $q$ satisfying the disparity known?

4. For a $p,q$ pair from (1), labelled so that there are more groups with order $p^k$ than $q^k$, does the same hold for all $\ell\geq k$, that there are more groups with order $p^\ell$ than $q^\ell$? I would think so, just by taking direct products, but it is conceivable that there is some hiccup that may cause $q$ to have a huge skip from $q^{\ell-1}$ to $q^\ell$ that $p$ did not enjoy, and THAT would be VERY interesting.

As an example of a difference between primes, a peculiarity to the prime $2$ is that a $2$-group whose exponent is $2$ is necessarily abelian while there are nonabelian $p$-groups of exponent $p$ when $p$ is odd, namely extraspecial $p$-groups. In my undergrad days, I set a task for seeing if "$2$" could be generalized to "prime" and quickly discovered how the proof broke down, but did not discover extraspecial $p$-groups, the basic counterexamples, until grad school. With that as backdrop, I am particularly interested in examples highlighting the adage that "2 is the oddest prime."

This is intentionally very flexible, so have fun with it!

Answer 1

If it exists, what is a value $k$ for which there are different numbers of (isomorphism classes of) groups of order $p^k$ and $q^k$?

You already observed that $k$ can't be $1$, $2$, or $3$. So such $k$ is at least $4$, and in fact $k = 4$ is an example: there are $14$ groups of order $2^4$ and $15$ groups of order $p^4$ for prime $p > 2$ (see here for $p > 2$). For groups of order $p^5$, the number of them depends on $p \bmod 12$: there are $51$ of order $2^5$, $67$ of order $3^5$, and for $p > 3$ the count is

$2p+71$ if $p \equiv 1 \bmod 12$,

$2p+67$ if $p \equiv 5 \bmod 12$,

$2p+69$ if $p \equiv 7 \bmod 12$,

$2p+65$ if $p \equiv 11 \bmod 12$.

Answer 2

Coclass theory for $p$-groups was initiated by Charles Leedham-Green and Mike Newman and collects many results for $p$-groups that look general but depend on $p$ specifically. For a quick summary of how it started, see the Wikipedia page on coclass and note that e.g. the solvable length of a $p$-group can be bounded in terms of $p$ and the coclass.

More recently, "coclass trees" (or "coclass graphs", "coclass forestgs", "descendant trees") have been studied, showing striking similarities and difference between the behavior for different primes. Once again the wikipedia page is not bad and also has some nice graphics.