Dihedral group, semi direct product, and Klein 4 subgroup

Question

Can someone help me out my confusion?

If the dihedral group $D_8$ (of order $8$) is the semi-direct product of $\mathbb{Z}_4$ and $\mathbb{Z}_2$, how is it possible that it contains the non-cyclic Klein subgroup $\mathbb{Z}_2 \times \mathbb{Z}_2$?

Thanks in advance.

Answer 1

Let $D_8=<a,b>=\{1,a,a^2,a^3,b,ab,a^2b,a^3b\}$ then you can see that $a^2\in Z(D_8)$ thus

$V=<a^2,b>$ is noncylic group order $4$, we are done.

By the way semidirect product does not preserve many the structure of a group,for example $S_3$ is semidirect product of two cyclic group $Z_2$ and $Z_3$ and it is nonabelian group with $Z(S_3)=1$.

Answer 2

A general observation is that if $G$ is the semidirect product $NH$ where $N \lhd G$ and $H \cap N = 1,$ then whenever $K$ is a characteristic subgroup of $N,$ $G$ will contain the semidirect product $KH.$The case of a dihedral group is a special case of this, with $H$ cyclic of order $2$,$N$ cyclic of order $4$ and $K$ the unique subgroup of $N$ of order $2.$ A characteristic subgroup of a group $G$ is a subgroup $X$ of $G$ such that $\alpha(X) = X$ for each automorphism $\alpha$ of $G.$