Minimal normal subgroup of a finite group.
I have Theorem 1 from a research paper.
Theorem 1. Suppose that is a finite non-abelian simple group. Then there exists an odd prime such that has no -Hall subgroup.
I have Theorem 2 from a book.
Theorem 2. If is a minimal normal subgroup of , then either is an elementary abelian -group for some prime or is the direct product of isomorphic nonabelian simple groups.
I like to know if the contradicition in this argument is true.
Let be a minimal normal subgroup of a finite group with divides the order of and is not a -group. Let be a Sylow -subgroup of . Suppose that for each prime dividing the order of , there exists a Sylow -subgroup of such that is a subgroup of . Cleary, for some postive intger , where each is a nonabelian simple group. Let be one of the . It is clear that if some prime divides the order of then this prime must divide the order of as . Since is normal in , then (I Know that this statement is true so do not bother checking it). So, is a finite nonableian simple group with Hall -subgroup for each odd prime dividing the order of which contradicts Theorem 1.
Thanks in advance.