Order of simple non-abelian group divides half the factorial of every Sylow number
- Order of simple non-Abelian group divides factorial of every Sylow number
- Order of simple non-Abelian group divides factorial of index of proper subgroup
- Simple non-Abelian group is isomorphic to subgroup of symmetric group on left coset space of proper subgroup
- Simple non-Abelian group is isomorphic to subgroup of alternating group on left coset space of proper subgroup of finite index
Related survey articles
Small-index subgroup technique: The use of this and similar results to prove that groups satisfying certain conditions (e.g., conditions on the order) cannot be simple.
- Sylow subgroups exist
- Prime power order implies not centerless
- Sylow number equals index of Sylow normalizer
- Order of simple non-Abelian group divides half the factorial of index of proper subgroup
Given: A finite simple non-Abelian group , a prime dividing the order of . is the number of -Sylow subgroups.
To prove: The order of divides .
- Let be a -Sylow subgroup of (such a exists by fact (1)).
- is not normal in : Since divides the order of , is nontrivial. Thus, is a nontrivial normal subgroup of . Also, cannot equal because then would be a group of prime power order, hence have a nontrivial center by fact (2), contradicting the assumption that it is simple non-Abelian.
- The subgroup is a proper subgroup of with index : The index being follows from fact (3). The proper part follows from the previous step, which concluded that is not normal in .
- The order of divides : This follows from the previous step and fact (4).