Why are regular $p$-groups called "regular?"

Question

In the concept of regular $p$-Groups, what does "regularity" stand for? What is "regular" in such groups?

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I would like to know idea behind defining these groups, and naming these groups "regular." I know that they carry many interesting properties of abelian groups, although they themselves are non-abelian. But, this may not be the idea of Philip Hall to introduce the regular $p$-groups.

Answer 1

I think the following might explain why Philip Hall chose the term regular, there is more to it than the fact that such $p$-groups have many nice properties. In the introduction of "A Contribution to the Theory of Groups of Prime Power Order", Philip Hall writes (pg. 33):

.. It results from the formula of §3 that any $p$-group of class less than $p$ is regular; and, in particular, that any group of order $p^n$ with $n \leq p$ is regular. Thus, if we confine our attention to the groups of order $p^n$ for some particular $n$, then only a finite number of these groups are irregular. In this sense, then, regular groups are the rule, irregular ones the exception, since they occur only for the first few values of $p$.

In other words, when we fix $n$ and study $p$-groups of order $p^n$ in general, there are first finitely many irregular cases to deal with, and the rest are regular. For example, with $n = 7$ there exist irregular $p$-groups of orders $2^7$, $3^7$ and $5^7$, but $p$-groups of order $p^7$ are regular when $p \geq 7$.