FiSGO.PIrrepsSearch.build_single_bound

build_single_bound(n)

Bounds on the prime powers of an n-dimensional (quasi)primitive group in characteristic zero.

The bounds are computed using the following results:

Let \(G\) be an n-dimensional (quasi)primitive group in characteristic zero. Let \(p\) be a prime and \(k\) denote the largest power of \(p\) dividing \(|G|\). Let \((n!)_p\) denote the largest power of \(p\) dividing \(n!\).

• (Brauer) if \(p \,|\, |G|\) then \(p < 2n+1\).

• (Brauer) if \(p > n+1\) then \(p^2\) does not divide \(|G|\).

• (Blichfeld, Brauer) if \(p\) is coprime to \(|G|\), then \(k < \log_p (n!)_p + n-1\).

• (Blichfeld) if \(p \,|\, |G|\), then \(p^k <= (n!)_p 6^{n-1}\).

Parameters:

n (int) – Dimension

Return type:

list[int]

Returns:

List of bounds to the prime powers dividing the order of the group. Each entry of the list corresponds to a prime, in ascending order starting from 2.