Hall and central factor implies direct factor

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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

Statement

Any Hall subgroup of a finite group that is a central factor of the group, is also a direct factor.

Facts used

  1. Second isomorphism theorem
  2. Normal Hall implies permutably complemented (this is the first half of the Schur-Zassenhaus theorem)

Proof

Given: A finite group G, a Hall subgroup H such that HC_G(H) = G.

To prove: H is a direct factor of G.

Proof: Consider the subgroup C_G(H). The subgroup Z(H) = H \cap C_G(H) is central in C_G(H). Also, by the second isomorphism theorem, C_G(H)/(H \cap C_G(H)) \cong HC_G(H)/H = G/H, so Z(H) is a Hall subgroup of C_G(H).

Thus, Z(H) is a central Hall subgroup of C_G(H).

In particular Z(H) is normal Hall in C_G(H), so it possesses a complement, say K, in C_G(H). Clearly, H and K permute element-wise, because K \le C_G(H), and H \cap K is trivial, because K \cap (H \cap C_G(H)) is trivial by construction. Finally, Z(H)K = C_G(H) by construction, so HK = H(Z(H)K) = HC_G(H) = G, and so, H and K are complements.

Thus: H and K commute element-wise, generate the whole group, and intersect trivially, making G an internal direct product of H and K, and thus making H a direct factor of G.