Linear Algebra

• Jordan Form

RelationShip between Jordan Block and Eigen Values

Characteristic Equation --> $ \lambda $ of the set of positive integers $n$ such that every group of order $n$ is (i) cyclic, (ii) abelian, or (iii) nilpotent.
Say that a positive integer $n > 1$ is a **nilpotent number** if $n = p_1^{a_1} \cdots p_r^{a_r}$ (here the $p_i$'s are distinct prime numbers) and for all $1 \leq i,j \leq r$ and $1 \leq k \leq a_i$, $p_i^k \not \equiv 1 \pmod{p_j}$. Also, let us say that $1$ is a nilpotent number.
(So, for instance, any prime power is a nilpotent number. A product of two distinct primes $pq$ is a nilpotent number unless $p \equiv 1 \pmod q$ or $q \equiv 1 \pmod p$.)
Then, for $n \in \mathbb{Z}^+$:

• (Pazderski, 1959) Every group of order $n$ is nilpotent iff $n$ is a nilpotent number.
• (Dickson, 1905) Every group of order $n$ is abelian iff $n$ is a cubefree nilpotent number.
• (Szele, 1947) Every group of order $n$ is cyclic iff $n$ is a squarefree nilpotent number. and n is cyclic iff $gcd(n,\phi(n))=1$

For example, if $n = pq$ is a product of distinct primes, then $n$ is squarefree, so every group of order $n$ is nilpotent iff every group of order $n$ is abelian iff every group of order $n$ is cyclic iff $p \not \equiv 1 \pmod q$ and $q \not \equiv 1 \pmod p$. In particular, every group of order $15$ is cyclic. Paper