Q: Given a fraction A/B when does there exist a finite group G and an automorphism f s.t. exactly A/B elements of G are mapped to their own inverses (f(a) = a(adsbygoogle = window.adsbygoogle || []).push({}); ^{-1}? If so how can we find the group? Does anything change if we allow infinite groups?

I have a friend who was preparing for an intro to abstract algebra class. So we were going over questions that the professor had asked him. On one exam he had been asked if there was a group with automorphism that sent 3/4 of the group to their own inverses. He found by considering small groups that D_{4}the symmetries of the circles worked with the identity automorphism. We could not think of a way to solve this question in general.

if 1 > A/B > 1/2 then G can't be abelian because in abelian group the elemnts mapped homomorphically to their own inverses is a subgroup.

If A/B = C/D * E/F then we can find the solution for the RHS groups we can take a direct product and get A/B.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Fraction of Elements in a group mapped to own inverse by automorphism

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

Loading...

Similar Threads for Fraction Elements group |
---|

I Spin group SU(2) and SO(3) |

I What is difference between transformations and automorphisms |

I Lorentz group representations |

I Correspondence Theorem for Groups ... Another Question ... |

**Physics Forums | Science Articles, Homework Help, Discussion**