# Translate this theorem from Endliche Gruppen

In a paper I was doing a reference is given from "Endliche gruppen" by Huppert. I do not understand german and google translator was also not much helpful. Can some translate this theorem or much better give a reference from some book in english where same theorem has been stated or proved in english. Thanks.

This is the theorem-

It translates as (at least mathematically. I have changed from the notation $x^y$ to $y(x)$ and changed to some more common letters for the various things):
a) Let $N\unlhd G$ and $\alpha$ be an automorphism of $G$ such that $\alpha(n) = n$ for all $n\in N$ and $\alpha(g)N = gN$ for all $g\in G$.
Then $\alpha(g) = gF(gN)$ for $1$-cocycle $F$ in $Z^1(G/N,Z(N))$.
Thus, the center $Z(N)$ of $N$ can be regarded as a right $G/N$-module via the obviously welldefined assignment $x(Ng) = xg$ for $x\in Z(N)$ and $g\in G$.
b) Let $F\in Z^1(G/N,Z(N))$. Then the map $\alpha$ with $\alpha(g) = gF(gN)$ is an automorphism of $G$ which fixes $N$ and $G/N$ point-wise.
c) Let the automorphism $\alpha$ and the $1$-cocycle $F$ in $Z^1(G/N,Z(N))$ from a) be fixed. Then $F\in B^1(G/N,Z(N))$ iff there is some $h\in N$ with $\alpha(g) = hgh^{-1}$ for all $g\in G$.

• Is there any other text in english which proves this? Jan 31, 2016 at 13:50
• Also can you tell what $B^1(G/N,Z(N))$ means here. Is it set of all 1-coboundaries? Jan 31, 2016 at 13:54
• There is almost certainly also some reference in English, but I don't know one off the top of my head. And yes, those are the coboundaries. Jan 31, 2016 at 14:34