Mathematics Asked by Benjamin on October 3, 2020

Let $G$ be a finite non-cyclic group. Can a non-inner automorphism map every subgroup to its conjugate? Namely, can there be a non-inner automorphism $alpha$ that, for every $Hle G$, there exists some $g$ in $G$ such that $alpha(H)=H^g$?

Yes, the dihedral group of order 10 has this property. Its subgroup structure is very simple: $D_{10}$, ${e}$, the rotation subgroup, and the 5 subgroups generated by a flip. Any automorphism fixes the first three, so those are done, and shuffles the last five, and those 5 subgroups are all conjugate to each other.

All we need now is to show that there is a non-inner automorphism, but this is easy; the inner automorphisms always send a rotation to its inverse (or fix it) so we only need an automorphism which doesn't do that. Let the generators be $sigma,tau$, rotation and flip, and consider the automorphism defined on the rotations by $sigma mapsto sigma^2$ and fixing $tau$.

Correct answer by TokenToucan on October 3, 2020

Get help from others!

Recent Questions

- How can I transform graph image into a tikzpicture LaTeX code?
- How Do I Get The Ifruit App Off Of Gta 5 / Grand Theft Auto 5
- Iv’e designed a space elevator using a series of lasers. do you know anybody i could submit the designs too that could manufacture the concept and put it to use
- Need help finding a book. Female OP protagonist, magic
- Why is the WWF pending games (“Your turn”) area replaced w/ a column of “Bonus & Reward”gift boxes?

Recent Answers

- haakon.io on Why fry rice before boiling?
- Jon Church on Why fry rice before boiling?
- Peter Machado on Why fry rice before boiling?
- Lex on Does Google Analytics track 404 page responses as valid page views?
- Joshua Engel on Why fry rice before boiling?

© 2024 TransWikia.com. All rights reserved. Sites we Love: PCI Database, UKBizDB, Menu Kuliner, Sharing RPP