# Splittable group

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

A group $G$ generated by proper subgroups $H$ and $K$ with $H$ normal in $G$ and $H \cap K = E$( so that the quotient group $G/H$ is isomorphic to $K$, cf. Normal subgroup). $G$ is called a split extension of the group $H$ by the group $K$, or a semi-direct product of $H$ and $K$. If the subgroups $H$ and $K$ commute elementwise, i.e. $hk = kh$ for all $h \in H$, $k \in K$, their semi-direct product coincides with the direct product $H \times K$. A semi-direct product $G$ of a group $H$ and a group $K$ is given by a homomorphism $\psi$ of $K$ into the group $\mathop{\rm Aut} H$ of automorphisms of $H$. In this case, the formula

$$( h _ {1} , k _ {1} ) ( h _ {2} , k _ {2} ) = \ ( h _ {1} \psi ( k _ {1} ) ( h _ {2} ) , k _ {1} k _ {2} )$$

for all $h _ {1} , h _ {2} \in H$, $k _ {1} , k _ {2} \in K$, defines the multiplication in $G$. In the case when $K = \mathop{\rm Aut} H$ and $\psi$ is the identity mapping, $G$ is called the holomorph of $H$( cf. Holomorph of a group).

#### References

 [1] D. Gorenstein, "Finite groups" , Chelsea, reprint (1980)

Conversely, if $G = HK$ is a semi-direct product, then conjugation with $k$ in $G$ defines a homomorphism $\psi : K \rightarrow \mathop{\rm Aut} H$ from which $G$ can be reconstructed, i.e.
$$\psi ( k) ( h) = k h k ^ {-} 1 .$$
As a set the semi-direct product of $H$ and $K$ is $H \times K$. The subsets $\{ {( h , 1) } : {h \in H } \}$, $\{ {( 1, k) } : {k \in K } \}$ are subgroups that identify with $H$ and $K$.