Lemma For a group $G$, suppose that $G = A \times B $. Then for a subset $C \subset G$, $G= C \times B \iff C = \{\alpha\cdot \phi(\alpha) \mid \alpha \in A\}$, where $\phi : A \mapsto \text{Cent}(B)$ is a homomorphism.
Question : Is there any way to rephrase the above lemma in simple plain english?What is its significance?
It means $C$ is a "tilted" copy of $A$ within $A\times B$.
Consider the case of $A$ the $y$-axis and $B$ the $x$-axis in $G=(\mathbb{R}^2,+)$. In order for $G=C\times B$ (let's say as vector spaces so nothing janky happens) you need $C$ to be another line through the origin that is not vertical, which will be the same as $A$ but "tilted" so that some of its elements $a\in A$ turn into elements $a+\phi(a)$. The function $\phi$ records exactly how $A$ is getting pushed to lean over $B$, and in order for the result to remain a subgroup, $\phi$ needs to be a linear map $\phi:A\to B$. (When you switch back to groups, then you need $\phi$ to be a homomorphism $A\to Z(B)$.)