Construct a group isomorphism $\phi : U(20) \to \mathbb Z_2 \oplus \mathbb Z_4$.

Question

Construct a group isomorphism $\phi : G_1 \to G_2$, with $G_1 = U(20)$ and $G_2 = \mathbb Z_2 \oplus \mathbb Z_4$.

EDIT: removed mapping because it was not an isomorphism

Answer 1

Hint: $\Bbb Z _2 \oplus\Bbb Z_4$ is generated by two elements, of orders $2$ and $4$.

details: define $$ F(1,0)=19, F(0,1)=17 $$ such as $F$ is an homomorphism. This is possible because the orders of $19$ and $17$ are 2 and 4.

You then get $$ F(0,2) = 17\times 17 = 9\\ F(0,3) = 17\times 9 = 13\\ F(1,1) = 19\times 17 = 3\\ F(1,2) = 19\times 9 = 11\\ F(1,3) = 19\times 13 = 7 $$ hence $F$ is an isomorphism.

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This was the naïve approach.

As soon as you found $a$ of order 4 and $b$ of order 2 and $b\neq a^2$, you know that the subgroup $\langle a\rangle$ is of order 4 and does not contain $b$.

So the subgroup $\langle a,b\rangle$ has no other choice than being $U(20)$.

Then the morphism has to be onto (without exhaustive check).