Core-nontrivial prime divisor

This article defines a property of a prime divisor of an integer, one that is important/useful in the use of Sylow theory to study groups

Definition

Symbol-free divisor

A prime divisor of a number is said to be core-nontrivial if for every group whose order is that number, the core for that prime divisor is nontrivial.

Definition with symbols

A prime divisor $p$ of a number $N$ is said to be core-nontrivial if for any group $G$ of order $N$, the group $O_p(G)$ (viz the $p$-core of $G$) is nontrivial.

Testing for core-nontriviality

The typical technique used here is the Sylow intersection technique. The idea works as follows. Suppose we know that a group has order $p^rq$ with $r > 1$. Then the claim is that $p$ is core-nontrivial unless $q \equiv 1 \mod p^2$. The rough sketch of the proof is as follows:

There are two cases, viz $n_p = 1$ and $n_p \ne 1$. In case <math.n_p = 1[/itex], thewhole Sylow subgroup is normal, hence it is the normal core, and the core is nontrivial.

Suppose there exist two Sylow subgroups $P$ and $Q$ such that $P \cap Q$ is of index $p$ in both. Then $P \cap Q$ is normal in both $P$ and $Q$, hence it is normal in $$. But since $P$ is a subgroup of prime index, it is maximal, and hence, $ = G$. Thus $P \cap Q$ is a normal $p$-subgroup of $G$. Since $r > 1$, $P \cap Q$ has size $p^{r-1} > 1$ and we have a nontrivial core.

The only other case is that any two Sylow subgroups intersect in a subgroup of index at least $p^2$ in both. But in that case, we can show that $n_p \equiv 1 \mod p^2$ (refer Sylow intersection technique). Since $n_p|q$ and $q$ is prime, this forces $q \equiv 1 \mod p^2$.

Thus, if $q$ is not 1 modulo $p^2$, $p$ is a core-nontrivial prime divisor.