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-

enter image description here


1 Answer 1


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 $g\in G$.

  • $\begingroup$ Is there any other text in english which proves this? $\endgroup$ Jan 31, 2016 at 13:50
  • $\begingroup$ Also can you tell what $B^1(G/N,Z(N))$ means here. Is it set of all 1-coboundaries? $\endgroup$ Jan 31, 2016 at 13:54
  • $\begingroup$ 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. $\endgroup$ Jan 31, 2016 at 14:34

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.