Core-nontrivial prime divisor

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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.

Relation with other properties

Stronger properties

Related 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 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</math>, 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 <P,Q>. But since P is a subgroup of prime index, it is maximal, and hence, <P,Q> = 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