Splittable group

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)

Comments

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 $.