I tried to prove that for any $n$ $\geq$ 3 there are exactly $4$ distinct homomorphisms from $Z_2$ into $Aut(Z_2^n)$. Also the resulting semidirect products give $4$ non isomorphic groups of order $2^{n+1}$. But I couldn't get any idea to start with.How can I get the best solution for both of these problems?
2026-03-27 20:21:07.1774642867
For any $n ≥3$ there are exactly $4$ distinct homomorphisms from $Z_2$ into $Aut(Z_2^n).$
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