Timeline for Automorphisms and epimorphisms of finite groups
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 28, 2019 at 15:16 | comment | added | Derek Holt | Note also that $K/N$ is a subdirect product of (an unfortunately large number of) copies of $G$, so if $G$ is abelian, nilpotent, solvable, etc, then so is $K/N$. | |
| Jun 28, 2019 at 14:48 | vote | accept | Neil Strickland | ||
| Jun 28, 2019 at 14:44 | comment | added | Neil Strickland | To spell it out, the fibers of $f$ and $g$ will all have the same size (namely $|G|/|H|$), so we can choose a bijection $h_1\colon G \to G$ (not a homomorphism) with $g=fh_1$. Then $h_1$ induces an automorphism of $K$, which preserves the characteristic subgroup $N$ and so induces an automorphism $h$ of $K/N$. This will satisfy $gp=fph$. | |
| Jun 28, 2019 at 13:05 | comment | added | Neil Strickland | I thought about this construction before. I could not see how to prove that it had the required property, but perhaps I do see that now. I will try to sort out the details. | |
| Jun 28, 2019 at 12:25 | history | answered | Derek Holt | CC BY-SA 4.0 |