# 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 $.

**How to Cite This Entry:**

Splittable group.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Splittable_group&oldid=55831