# When is $A\rtimes_{\phi_1} B \cong A\rtimes_{\phi_2} B$?

Suppose $$1\to K \stackrel{m}{\rightarrow} G \stackrel{f}{\rightarrow} H \to 1$$ is short exact sequence of groups.

The followings are equivalent:

$$(1)\ G\cong K \times H;$$

$$(2)$$ The sequence right splits (i.e. $$\exists$$ homomorphism $$g:H \to G$$ s.t. $$f\circ g =$$ Id$$_H$$) and $$H\cong N \triangleleft G;$$

$$(3)\ G$$ is semidirect product of $$K$$ and $$H$$, and $$H$$ acts on $$K$$ trivially.

$$(4)$$ The sequence left splits (i.e. $$\exists$$ homomorphism $$h:G \to K$$ s.t. $$h\circ m =$$ Id$$_K$$).

However, if $$H$$ acts on $$K$$ nontrivially, $$G$$ may also be direct product.

e.g. let $$G$$ be a nonabelian group with $$h \in G\backslash Z(G)$$.

for $$1\to G \to G\times \Bbb Z \to \Bbb Z \to 1$$, splitting map $$g: \Bbb Z \to G \times \Bbb Z, 1 \mapsto (h,1).$$

$$\phi: \Bbb Z \to \text{Aut} G,\ \phi(1)(g,1)=(h,1)(g,1)(h,1)^{-1}=(hgh^{-1},1).$$

Since $$h \not \in Z(G)$$, this action is nontrivial.

So under what condition, semidirect product of groups is isomorphic to their direct products?

And more generally, when is $$A\rtimes_{\phi_1} B \cong A\rtimes_{\phi_2} B$$?

• What exactly are you asking? As far as I can see everything you have written is correct, and there is no contradiction. It is possible for a direct product $K \times H$ to be isomorphic to a semidirect product $K \rtimes H$ with $H$ acting nontrivially on $K$. A smaller example is $K=S_3$, $H=C_2$, where the nontrivial action of $H$ on $K$ is an inner automorphism. – Derek Holt Feb 9 at 15:23
• "when is $A\rtimes_{\phi_1} B \cong A\rtimes_{\phi_2} B$?" AFAIK there is no general rule and these have to be considered case by case. You may also want to read this: math.stackexchange.com/questions/527800/… – freakish Feb 14 at 12:34

## 1 Answer

There're a number of questions and answers for this problem in MSE (as you can see on the right), and I'd like to make a summary.

For groups $$N,H$$, group homomorphism $$\varphi,\phi:H\to \text{Aut }N$$ represents an action of $$H$$ on $$N$$.

$$(1)$$ If $$\exists f\in\text{Aut}(H),g\in \text{Aut}(N)$$ s.t. $$\varphi\circ f=c_g \circ \phi$$, where $$c_g \circ\phi(h)=g\phi(h)g^{-1}$$,then

$$N\rtimes_\phi H\cong N \rtimes_\phi H$$ via $$\psi:(n,h)\mapsto(g(n),f(h))$$.

$$(2)$$ If im$$\varphi\subset \text{Inn }N$$, then $$G=NC_G(N).$$ And since $$C_G(N)\triangleleft G$$, if in addition $$Z(N)=1$$

i.e. $$N\cap C_G(N)=1,$$ then $$G\cong N\times C_G(N)$$ and $$G/N\cong C_G(N)$$.

From $$G/N \cong H, H \cong C_G(N)$$ and we have $$N\rtimes_\varphi H\cong G\cong N \times C_G(N)\cong N \times H.$$