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Given $n = \lvert G \rvert$ then the factorization of $n$ can sometimes give information about the group $G$. For example if $n$ is prime we know that the group is the cyclic group $C_n$. If $n$ is the power of a prime then we know that $G$ is nilpotent. Are there cases like these that assert that $G$ has certain properties knowing $n$? For example all groups of size $100$ are solvable, for which $n$ can one say the same?

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    Have you heard about the [Sylow theorems](https://en.m.wikipedia.org/wiki/Sylow_theorems)? – 2017-02-04
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    What properties to they tell us about the structure of the group apart from the existence of some subgroups? – 2017-02-04
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    @Arthur In the case $n=100$ they tell us that they have a normal subgroup of order $25$; – 2017-02-04
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    Existence and number of certain subgroups is an important part of a group's structure. – 2017-02-04
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    Do you mean normal subgroups or ore there other importtant cases? – 2017-02-04
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    I mean other cases too. It doesn't directly help you make a multiplication table, but it _is_ important. – 2017-02-04

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We say that $n$ is a nilpotent number if when we factor $n = p_1^{a_1} \cdots p_r^{a_r}$ we have $p_i^k \not \equiv 1 \bmod{p_j}$ for all $1 \leq k \leq a_i$.

Then

  • Every group of order $n$ is nilpotent iff $n$ is a nilpotent number.
  • Every group of order $n$ is abelian iff $n$ is a cubefree nilpotent number.
  • Every group of order $n$ is cyclic iff $n$ is a squarefree nilpotent number.

(adapted from an answer by Pete Clark)

For solvable numbers, see the last theorem in the paper Nilpotent and solvable numbers by Pakianathan and Shankar:

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(also J. Pakianathan and K. Shankar, Nilpotent numbers, Amer. Math. Monthly, 107, August– September 2000, 631–634)