Semidirect product of two groups defined in terms of a homomorphism

Question

I am going through From optimal measurement to efficient quantum algorithms for the hidden subgroup problem over semidirect product groups by Bacon et al.

I am having trouble to understand the definition of semidirect product in section 2.2. Here is the screenshot.

My questions:

1. Shouldn't the relation be $(a, b) (a', b') = (a + a', b+b')$ instead of being $(a, b) (a', b') = (a + \varphi(b)(a'), b+b')$?
2. What is the significance of $\varphi(b)(a')$ in this context?
3. I also try to verify the group inversion i.e. whether $(a, b) + (\varphi(-b)(-a), -b) = 0$.

$$(a, b) + (\varphi(-b)(-a), -b)$$ $$\implies (a + \varphi(-b)(-a), b -b)$$

I do not know how to proceed further from here.

Answer 1

1. When $\varphi$ is the identity, the semi-direct product coincides with the direct product.

2. It generalizes the notion of direct product.

3.

\begin{eqnarray*} (a,b)+(\varphi(-b)(-a),-b)&=&(a+\varphi(b)\varphi(-b)(-a),0)\\ &=&(a+\varphi(b-b)(-a),0)\\ &=&(a+\varphi(0)(-a),0)\\ &=&(a-a,0)\\ &=&(0,0) \end{eqnarray*}