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

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SUMMARY

The discussion centers on the existence of finite groups G and automorphisms f that map a specific fraction A/B of elements to their own inverses. It was established that for fractions where 1 > A/B > 1/2, G cannot be abelian, as abelian groups map elements homomorphically to their own inverses, forming a subgroup. The example of the dihedral group D4 was highlighted as a case where the identity automorphism achieves this mapping. Additionally, the discussion suggests that if A/B can be expressed as a product of fractions, solutions can be derived through direct products of groups, with references to Galois groups of field extensions for further exploration.

PREREQUISITES
  • Understanding of group theory concepts, particularly automorphisms.
  • Familiarity with finite and infinite groups.
  • Knowledge of abelian groups and their properties.
  • Basic comprehension of Galois theory and field extensions.
NEXT STEPS
  • Research the properties of dihedral groups, specifically D4, and their automorphisms.
  • Study the implications of automorphisms in abelian groups and their subgroup structures.
  • Explore Galois groups and their role in field extensions related to group theory.
  • Investigate direct products of groups and how they can be used to construct new groups with specific properties.
USEFUL FOR

This discussion is beneficial for students of abstract algebra, mathematicians exploring group theory, and researchers interested in the properties of automorphisms in both finite and infinite groups.

deluks917
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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-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 D4 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.
 
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I would consider Galois groups of field extensions. It should be possible to find extension towers with the required properties, depending a bit on ##A## and ##B##.
 

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