Let and be subgroups of a group satisfying the following two properties.
(i) All elements of can be written in the form for some and some .
Prove that and are normal subgroups of if and only if all elements of commute with all elements of .
State and prove Cauchy's Theorem.
Let and be distinct primes. Prove that an abelian group of order is isomorphic to . Is it true that all abelian groups of order are isomorphic to ?