$U(2^n) \thickapprox Z_2\oplus Z_{2^{n-2}}$ Show that $$U(2^n) \thickapprox Z_2\oplus Z_{2^{n-2}}\,\,\,\,\,\,,\text{for }n\geq3$$

Well I think this can be done with the help of this theorem
Let $m=n_1n_2.\,.\,.\,.n_k$ where $\gcd(n_i,n_j)=1$ for $i\neq j.$ Then,
$$U(m) \thickapprox U(n_1)\,\oplus U(n_2)\, \oplus .\,.\,.\,.\oplus\, U(n_k) $$

The theorem you are quoting does not help. Instead, show that 


*

*$-1$ has order $2$ in $U(2^{n})$,

*$5$ has order $2^{n-2}$, 

*the subgroup generated by $5$ does not contain $-1$ .

