Hol(G) As A Semi-direct Product
If is the automorphism group of then
where the multiplication is given by
Typically, a semidirect product is given in the form where and are groups and is a homomorphism and where the multiplication of elements in the semi-direct product is given as
which is well defined, since and therefore .
For the holomorph, and is the identity map, as such we suppress writing explicitly in the multiplication given in above.
For example,
- the cyclic group of order 3
- where
- with the multiplication given by
- where the exponents of are taken mod 3 and those of mod 2.
Observe, for example
and note also that this group is not abelian, as, so that is a non-abelian group of order 6 which, by basic group theory, must be isomorphic to the symmetric group .
Read more about this topic: Holomorph (mathematics)
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