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Dec 15, 2021 at 15:18 history closed Yemon Choi
Carl-Fredrik Nyberg Brodda
Eric Peterson
Bugs Bunny
coudy
Needs more focus
Nov 8, 2021 at 10:56 comment added YCor @piper1967 I don't have a reference, but this seems to be a straightforward verification. In general, if $N$ is this normal subgroup, the quotient $K/\langle\!\langle h\rangle\!\rangle$ is naturally isomorphic to the finite group $G/N$.
Nov 8, 2021 at 10:44 comment added piper1967 @YCor can you give reference to the proof of the fact that the condition holds if and only if the normal subgroup of G generated by $\{g^{-1}u(g) : g \in G\}$ is G itself?
Nov 8, 2021 at 10:18 review Close votes
Dec 15, 2021 at 15:18
Nov 8, 2021 at 10:14 comment added Derek Holt The problem I always have with this type of question is that I have no idea what sort of answer the poster is looking for. All you can really say is that sometimes it is and sometimes it is not.
Nov 8, 2021 at 9:58 comment added YCor Yes. There are easy examples. (1) Take $G$ simple non-abelian and any non-trivial action for the semidirect product. (2) Take $G$ abelian of odd order and $h$ acting on $G$ by $x\mapsto -x$. Etc. In general let $u$ be the automorphism of $G$ induced by $h$. Then the required property holds iff the normal subgroup of $G$ generated by $\{g^{-1}u(g):g\in G\}$ is $G$ itself.
Nov 8, 2021 at 9:36 comment added HJRW I've taken the liberty of making the title more descriptive. Please feel free to edit as you please, but something much more speicific than the original title would be preferable.
Nov 8, 2021 at 9:35 history edited HJRW CC BY-SA 4.0
Made the title more descriptive
Nov 8, 2021 at 8:51 history edited Mark Wildon CC BY-SA 4.0
added 78 characters in body
Nov 8, 2021 at 8:43 comment added piper1967 (h) is a cyclic subgroup of K. I asked when K/H, where H is the normal subgroup generated by (h), is trivial.
Nov 8, 2021 at 8:37 comment added LeechLattice Does "$K/(h)$ trivial" implies K is infinite cyclic? If it's true, how could $G<K$ be normal?
Nov 8, 2021 at 8:22 history asked piper1967 CC BY-SA 4.0