Paper 3, Section II, D

Groups | Part IA, 2019

Let HH and KK be subgroups of a group GG satisfying the following two properties.

(i) All elements of GG can be written in the form hkh k for some hHh \in H and some kKk \in K.

(ii) HK={e}H \cap K=\{e\}.

Prove that HH and KK are normal subgroups of GG if and only if all elements of HH commute with all elements of KK.

State and prove Cauchy's Theorem.

Let pp and qq be distinct primes. Prove that an abelian group of order pqp q is isomorphic to Cp×CqC_{p} \times C_{q}. Is it true that all abelian groups of order p2p^{2} are isomorphic to Cp×CpC_{p} \times C_{p} ?

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