Core-nontrivial prime divisor

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.

Stronger properties

 * Sylow-unique prime divisor

Related properties

 * Closure-proper prime divisor

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 $$. But since $$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.

Presentations/talks on this

 * Using Sylow theory in the classification of finite simple groups