Is this a special semidirect product?

Question

Currently I am going through Representation theory of semi-direct products by Reyes. At the begining of the article, the author defines the semidirect product as follows.

$G = H \cdot B$ is a semidirect product where $H$ is a normal subgroup of $G$ and $B$ is isomorphic with a group of automorphisms of $H$.

This doesn't match with the definition given here which is:

Semidirect products.

We say that $G$ is (isomorphic to) a semidirect product of $M$ by $N$ if and only if there exist subgroups $H$ and $K$ of $G$ such that:

• $H\cong M$ and $K\cong N$;
• $H\triangleleft G$;
• $H\cap K=\{e\}$;
• $G=HK$.

Because, in the second definition $H$ is not required to be isomophic with a group of automorphism of $K$. So, can I say that the first definition is not a general definition of semidirect product?

Answer 1

See "outer" and "inner" semi-direct products in https://en.wikipedia.org/wiki/Semidirect_product