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Alireza Abdollahi
Alireza Abdollahi
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History of a conjecture/problem non-inner automorphisms of of order p in finite p-groups

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Max Horn
Max Horn
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The following conjecture/problem posed in The Kourovka Notebook in 1973 by Ya. G. Berkovich:

Problem 4.13. Prove that every finite non-abelian $p$-group admits an automorphism of order $p$ which is not an inner one. ($p$ as usual denotes a prime number)

I would like to know if anybody knows anything concerning history of this problem.

What I know are as follows: W. GashutzGaschütz has proved in 1966 that every finite $p$-group of order greater than $p$ has a non-inner automorphism of $p$-power order. One year before GashutzGaschütz, H. Liebeck has proved problem 4.13 for $p$-groups of class $2$ whenever $p>2$. The problem in its own is of course interstinginteresting, but I think maybe the main motivation to propose the roblemproblem is to strengthen Gashutz'sGaschütz's famous result, am I right?

What I really look for is to know: does the problem come from other mathematics problems that people already noted to and they stated it? For example, has it any relation to the classification of finite simple groups?!

I am Sorrysorry if you feel my questions are so vague! and I apologize in advance for.

The following conjecture/problem posed in The Kourovka Notebook in 1973 by Ya. G. Berkovich:

Problem 4.13. Prove that every finite non-abelian $p$-group admits an automorphism of order $p$ which is not an inner one. ($p$ as usual denotes a prime number)

I would like to know if anybody knows anything concerning history of this problem.

What I know are as follows: W. Gashutz has proved in 1966 that every finite $p$-group of order greater than $p$ has a non-inner automorphism of $p$-power order. One year before Gashutz, H. Liebeck has proved problem 4.13 for $p$-groups of class $2$ whenever $p>2$. The problem in its own is of course intersting, but I think maybe the main motivation to propose the roblem is to strengthen Gashutz's famous result, am I right?

What I really look for is to know: does the problem come from other mathematics problems that people already noted to and they stated it? For example, has it any relation to the classification of finite simple groups?!

I am Sorry if you feel my questions are so vague! and I apologize in advance for.

The following conjecture/problem posed in The Kourovka Notebook in 1973 by Ya. G. Berkovich:

Problem 4.13. Prove that every finite non-abelian $p$-group admits an automorphism of order $p$ which is not an inner one. ($p$ as usual denotes a prime number)

I would like to know if anybody knows anything concerning history of this problem.

What I know are as follows: W. Gaschütz has proved in 1966 that every finite $p$-group of order greater than $p$ has a non-inner automorphism of $p$-power order. One year before Gaschütz, H. Liebeck has proved problem 4.13 for $p$-groups of class $2$ whenever $p>2$. The problem in its own is of course interesting, but I think maybe the main motivation to propose the problem is to strengthen Gaschütz's famous result, am I right?

What I really look for is to know: does the problem come from other mathematics problems that people already noted to and they stated it? For example, has it any relation to the classification of finite simple groups?!

I am sorry if you feel my questions are so vague! and I apologize in advance for.

Source Link
Alireza Abdollahi
Alireza Abdollahi
  • 3.7k
  • 19
  • 27

History of a conjecture/problem non-inner automorphisms of of order p in finite p-groups

The following conjecture/problem posed in The Kourovka Notebook in 1973 by Ya. G. Berkovich:

Problem 4.13. Prove that every finite non-abelian $p$-group admits an automorphism of order $p$ which is not an inner one. ($p$ as usual denotes a prime number)

I would like to know if anybody knows anything concerning history of this problem.

What I know are as follows: W. Gashutz has proved in 1966 that every finite $p$-group of order greater than $p$ has a non-inner automorphism of $p$-power order. One year before Gashutz, H. Liebeck has proved problem 4.13 for $p$-groups of class $2$ whenever $p>2$. The problem in its own is of course intersting, but I think maybe the main motivation to propose the roblem is to strengthen Gashutz's famous result, am I right?

What I really look for is to know: does the problem come from other mathematics problems that people already noted to and they stated it? For example, has it any relation to the classification of finite simple groups?!

I am Sorry if you feel my questions are so vague! and I apologize in advance for.