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
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 of a number is said to be core-nontrivial if for any group of order , the group (viz the -core of ) is nontrivial.
Relation with other properties
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 with . Then the claim is that is core-nontrivial unless . The rough sketch of the proof is as follows:
There are two cases, viz and . In case <math.n_p = 1</math>, thewhole Sylow subgroup is normal, hence it is the normal core, and the core is nontrivial.
Suppose there exist two Sylow subgroups and such that is of index in both. Then is normal in both and , hence it is normal in . But since is a subgroup of prime index, it is maximal, and hence, . Thus is a normal -subgroup of . Since , has size and we have a nontrivial core.
The only other case is that any two Sylow subgroups intersect in a subgroup of index at least in both. But in that case, we can show that (refer Sylow intersection technique). Since and is prime, this forces .
Thus, if is not 1 modulo , is a core-nontrivial prime divisor.