P-extensible implies inner: Difference between revisions

Latest revision as of 16:27, 5 November 2009

Statement

Suppose $p$ is a prime number and $P$ is a p-group, i.e., a group in which the order of every element is a power of $p$ . Suppose $σ$ is an automorphism of $P$ such that for any $p$ -group $Q$ containing $P$ , there is an automorphism $σ^{'}$ of $Q$ whose restriction to $P$ equals $σ$ . Then, $σ$ is an Inner automorphism (?) of $P$ .

References

Journal references

Characterizing inner automorphisms of groups by Martin R. Pettet, Archiv der Mathematik, ISSN 1420-8938 (Online), ISSN 0003-889X (Print), Volume 55,Number 5, Page 422 - 428(Year 1990): ^{Springerlink official copy}^{More info}

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 ==Statement==
-Suppose <math>p</math> is a [[prime number]] and <math>P</math> is a [[p-group]], i.e., a group in which the order of every element is a power of <math>p</math>. Suppose <math>\sigma</math> is an [[automorphism]] of <math>P</math> such that for any <math>p</math>-group <math>Q</math> containing <math>P</math>, there is an automorphism <math>\sigma'</math> of <math>Q</math> whose restriction to <math>P</math> equals <math>\sigma</math>. Then, <math>\sigma</math> is an inner automorphism of <math>P</math>.
+Suppose <math>p</math> is a [[prime number]] and <math>P</math> is a [[p-group]], i.e., a group in which the order of every element is a power of <math>p</math>. Suppose <math>\sigma</math> is an [[automorphism]] of <math>P</math> such that for any <math>p</math>-group <math>Q</math> containing <math>P</math>, there is an automorphism <math>\sigma'</math> of <math>Q</math> whose restriction to <math>P</math> equals <math>\sigma</math>. Then, <math>\sigma</math> is an [[fact about::inner automorphism]] of <math>P</math>.
 ==References==