Let $M=\langle u,v: u^2=v^8=1, vu=uv^5\rangle$ be a group of order $16$.
I have a reading, that says:
Let $Z_2=\langle x\rangle$ and $Z_4=\langle y\rangle$, then $\varphi:Z_2\times Z_4\to M$ given by $(x^a,y^b)\to u^av^{2b}$ is an injective homomorphism.
Then they go on to show that $\varphi$ is indeed well-defined, and is injective, but I have real problem with their homomorphism proof: It goes like
Let $(x^{a_1},y^{b_1}), (x^{a_2},y^{b_2}) \in Z_2 \times Z_4.$ Then $\varphi((x^{a_1},y^{b_1}) (x^{a_2},y^{b_2})) = \varphi(x^{a_1+a_2}, y^{b_1+b_2}) = u^{a_1+a_2} v^{2(b_1+b_2)} = u^{a_1}v^{2b_1}u^{a_2}v^{2b_2} = \varphi(x^{a_1},y^{b_1}) \varphi(x^{a_2},y^{b_2})$
I can't get how do we get $u^{a_1+a_2} v^{2(b_1+b_2)} = u^{a_1}v^{2b_1}u^{a_2}v^{2b_2}$.
$uv^2u=(uvu)^2=v^2$ so $u$ and $v^2$ commute.