As summarized in the title, suppose there is an isomorphism between $G\_1 \times G\_2$ and $G\_3$, is it always true that $BG\_1 \times BG\_2$ is homotopy equivalent to $BG\_3$? If it is not always true could you give a counterexample and the condition for it to be true?


Similarly suppose there is an isomorphism between $G\_1 \rtimes G\_2$ and $G\_3$, is it always true that $BG\_1 \times BG\_2$ is homotopy equivalent to $BG\_3$? If it is not always true could you give a counterexample and the condition for it to be true?



$G_1 \rtimes G_2 \cong G_3 \implies BG_1 \times BG_2 \simeq BG_3$? $G_1 \times G_2 \cong G_3 \implies BG_1 \times BG_2 \simeq BG_3$?
Let $p\_{G\_i}:EG\_i\rightarrow BG\_i$ be the universal fibration, $i=1,2$, then
$p\_{G\_1}\times p\_{G\_2}:EG\_1 \times EG\_2 \rightarrow BG\_1\times BG\_2$ is the universal fibration of $G\_1\times G\_2$ since it is a $G\_1\times G\_2$ principal bundle and $EG\_1\times EG\_2$ is contractible, thus $BG\_1\times BG\_2=B(G\_1\times G\_2)$.


if you consider the semi-direct product $H$ of $G\_1$ and $G\_2$, since $G\_1$ is a subgroup of $H$, the quotient $EH/G\_1$ is a classifying space of $G\_1$. There is a section $s:G\_2\rightarrow H$, remark that $G\_2$ acts on $EH/G\_1$ via this section since $g\_2g\_1x=g\_2g\_1g\_2^{-1}g\_2x$, thus if $x=g\_1x, g\_1\in G\_1$, $g\_2(g\_1x)=g'\_1g\_2x$ since $G\_1$ is a normal subgroup, $(EH/G\_1)/G\_2=B(G\_1\times G\_2)$ you have a fibration $EH/G\_1 = BG\_1\rightarrow B(G\_1\times G\_2)$. 


