Paper 3, Section I, D

Groups | Part IA, 2013

State Lagrange's Theorem.

Let GG be a finite group, and HH and KK two subgroups of GG such that

(i) the orders of HH and KK are coprime;

(ii) every element of GG may be written as a product hkh k, with hHh \in H and kKk \in K;

(iii) both HH and KK are normal subgroups of GG.

Prove that GG is isomorphic to H×KH \times K.

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