Finite-quotient-pullbackable implies class-preserving: Difference between revisions

Revision as of 20:05, 8 April 2009

This article gives the statement and possibly, proof, of an implication relation between two automorphism properties. That is, it states that every automorphism satisfying the first automorphism property (i.e., finite-quotient-pullbackable automorphism) must also satisfy the second automorphism property (i.e., class-preserving automorphism)
View all automorphism property implications | View all automorphism property non-implications
Get more facts about finite-quotient-pullbackable automorphism|Get more facts about class-preserving automorphism

Statement

Suppose $G$ is a finite group and $\sigma$ is a finite-quotient-pullbackable automorphism of $G$ . Then, $\sigma$ is a class-preserving automorphism of $G$ : it sends every element of $G$ to within its conjugacy class.

Facts used

Finite-quotient-pullbackable implies Hall-quotient-pullbackable
Hall-quotient-pullbackable implies linearly pushforwardable over prime field when the prime does not divide the order of the group.
Linearly pushforwardable implies class-preserving when the field is a class-separating field
Every finite group admits a sufficiently large field
Sufficiently large implies splitting, Splitting implies character-separating, Character-separating implies class-separating

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 Suppose <math>G</math> is a [[finite group]] and <math>\sigma</math> is a [[finite-quotient-pullbackable automorphism]] of <math>G</math>. Then, <math>\sigma</math> is a [[class-preserving automorphism]] of <math>G</math>: it sends every element of <math>G</math> to within its [[conjugacy class]].
+==Facts used==
+# [[uses::Finite-quotient-pullbackable implies Hall-quotient-pullbackable]]
+# [[uses::Hall-quotient-pullbackable implies linearly pushforwardable over prime field]] when the prime does not divide the order of the group.
+# [[uses::Linearly pushforwardable implies class-preserving]] when the field is a [[class-separating field]]
+# [[uses::Every finite group admits a sufficiently large field]]
+# [[uses::Sufficiently large implies splitting]], [[uses::Splitting implies character-separating]], [[uses::Character-separating implies class-separating]]