What does the bar mean in $M = G \wr H = B \bar{H}$?

Question

Currently, I am going through Chapter 25 of Character theory of finite groups by Bertram Huppert.

On page $343$, there is a lemma on semidirect products. It starts as follows.

25.5 Lemma. Let $M = G \wr H = B \bar{H}$ be the wreath-product as described in 25.1.

My question:

• What is the meaning of $\bar{H}$?

Answer 1

I just found the answer on page 338.

Let me quote it here.

$\alpha$ with $(f, h) \alpha = h$ is an epimorphism of $G \wr H$ onto $H$ and

$$ \ker \alpha = B = \{(f, 1) | f: \Omega \to G\} $$

is isomorphic t $G \times \ldots \times G$. $B$ has a complement

$$ \bar{H} = \{(e, h) | h \in H\} $$

in $G \wr H$, where $e(j) = 1$ for all $j \in \Omega$ and $\bar{H} \cong H$ .