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Homomorphism and inequality

Let \mathcal{G}be a group and f:\mathcal{G} \rightarrow \mathcal{G} be a homomorphism. Prove that

    \[\left | f \left ( \mathcal{G} \right ) \right |^2 \leq \left | \mathcal{G} \right | \left | f \left ( f \left ( \mathcal{G} \right ) \right ) \right |\]

Solution

In general it holds that  |\mathcal{G}|=|\ker f| |f(\mathcal{G})| ( first isomorphism theorem ) . Taking that for granted we also have

    \[|f(\mathcal{G})| = \left|\ker f|_{f(\mathcal{G})} \right| |f(f(\mathcal{G}))|\]

and the inequality is equivelant to  \left|\ker f|_{f(\mathcal{G})} \right| \leqslant |\ker f| which is obviously true.

The exercise can also be found at mathematica.gr .

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