undergraduate math about permutation

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for example: Let f : G −→ H denote an isomorphism between the two

groups G,H.

(a) For any g ∈ G show that order(g)=order(f(g)).

(b) G is abelian iff H is abelian.

(c) G is cyclic iff H is cyclic.

(d) G and H have the same number of elements

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