Holomorph (mathematics) - Hol(G) As A Semi-direct Product

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 .