# Finite Groups of Local Characteristic p

# The P^{~}! Theorem

## Journal of Algebra, Volume 292, Issue 2, 15 October 2005, Pages 363-392

## Abstract:

In this paper we assume P!, Q!, gb>1 for P and that there exists
two minimal parabolics P_{1} and P_{2} containing S such
that M_{i}:=< P , P_{i} >
is contained in a local subgroup and P_{i} does not normalizes
P° . We show that p=3 or 5, M_{i}/Op(M_{i}) is isomorphic
to SL_{3}(p) and M_{i} has exactly 2 composition factors
in Op(M_{i}), namely two natural modules dual to each other.

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Last Revised 8/23/05