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If-G-1-and-G-2-are-groups-and-f-G-1-G-2-is-a-group-homomorphism-then-prove-that-o-G-1-o-G-2-




Question Number 11372 by agni5 last updated on 22/Mar/17
If G_1  and G_2  are groups , and f : G_1  →G_2   is a group homomorphism , then prove  that o(G_1 ) = o(G_2 ) .
$$\mathrm{If}\:\mathrm{G}_{\mathrm{1}} \:\mathrm{and}\:\mathrm{G}_{\mathrm{2}} \:\mathrm{are}\:\mathrm{groups}\:,\:\mathrm{and}\:\mathrm{f}\::\:\mathrm{G}_{\mathrm{1}} \:\rightarrow\mathrm{G}_{\mathrm{2}} \\ $$$$\mathrm{is}\:\mathrm{a}\:\mathrm{group}\:\mathrm{homomorphism}\:,\:\mathrm{then}\:\mathrm{prove} \\ $$$$\mathrm{that}\:\mathrm{o}\left(\mathrm{G}_{\mathrm{1}} \right)\:=\:\mathrm{o}\left(\mathrm{G}_{\mathrm{2}} \right)\:. \\ $$

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