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

In summary, the conversation discusses the existence of a finite group with an automorphism that maps exactly A/B elements to their own inverses. It is possible to find such a group using a direct product of smaller groups, but if A/B is between 1 and 1/2, the group cannot be abelian. It is also possible to consider Galois groups of field extensions to find a suitable group.
  • #1
deluks917
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4
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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  • #2
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##.
 

1. What is an automorphism?

An automorphism is a mathematical function that maps a structure onto itself, preserving the structure and properties of the original object.

2. What does "fraction of elements" refer to in this context?

In this context, "fraction of elements" refers to the proportion or percentage of elements in a particular group or set that have a certain property or relationship.

3. How is the inverse of an element defined in automorphism?

The inverse of an element in automorphism is defined as the element that, when composed with the original element, produces the identity element. In other words, it "undoes" the original element's operation.

4. Can you give an example of mapping elements to their own inverse by automorphism?

One example of this is the mapping of real numbers to their additive inverse (i.e. the negative of the original number) in the group of real numbers under addition. For example, the automorphism would map 5 to -5, 2 to -2, and so on.

5. Why is understanding the fraction of elements mapped to their own inverse by automorphism important?

Understanding this fraction can provide insight into the structure and symmetry of a mathematical object or system. It can also be useful in solving complex problems and developing new mathematical concepts and theories.

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