Hall and central factor implies direct factor
This article gives a proof/explanation of the equivalence of multiple definitions for the term Hall direct factor
View a complete list of pages giving proofs of equivalence of definitions
- Second isomorphism theorem
- Normal Hall implies permutably complemented (this is the first half of the Schur-Zassenhaus theorem)
Given: A finite group , a Hall subgroup such that .
To prove: is a direct factor of .
Proof: Consider the subgroup . The subgroup is central in . Also, by the second isomorphism theorem, , so is a Hall subgroup of .
Thus, is a central Hall subgroup of .
In particular is normal Hall in , so it possesses a complement, say , in . Clearly, and permute element-wise, because , and is trivial, because is trivial by construction. Finally, by construction, so , and so, and are complements.
Thus: and commute element-wise, generate the whole group, and intersect trivially, making an internal direct product of and , and thus making a direct factor of .