Class-preserving automorphism
This article defines a term that has been used or referenced in a journal article or standard publication, but may not be generally accepted by the mathematical community as a standard term.[SHOW MORE]
This article defines an automorphism property, viz a property of group automorphisms. Hence, it also defines a function property (property of functions from a group to itself)
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This is a variation of inner automorphism|Find other variations of inner automorphism |
Definition
An automorphism of a group is termed a class-preserving automorphism or class automorphism if it takes each element to within its conjugacy class. In symbols, an automorphism of a group is termed a class automorphism or class-preserving automorphism if for every in , there exists an element such that . The choice of may depend on .
Relation with other properties
Stronger properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
Inner automorphism | conjugation by an element | inner implies class-preserving | class-preserving not implies inner | Hall-extensible automorphism, Locally inner automorphism|FULL LIST, MORE INFO |
Locally inner automorphism | preserves conjugacy classes of finite tuples | locally inner implies class-preserving | class-preserving not implies locally inner | |FULL LIST, MORE INFO |
Weaker properties
Related properties
- Subgroup-conjugating automorphism: Further information: Class-preserving not implies subgroup-conjugating, Subgroup-conjugating not implies class-preserving
- Class-inverting automorphism
Facts
Metaproperties
Group-closedness
This automorphism property is group-closed: it is closed under the group operations on automorphisms (composition, inversion and the identity map). It follows that the subgroup comprising automorphisms with this property, is a normal subgroup of the automorphism group
View a complete list of group-closed automorphism properties
Clearly, a product of class automorphisms is a class automorphism, and the inverse of a class automorphism is a class automorphism. Thus, the class automorphisms form a group which sits as a subgroup of the automorphism group. Moreover, this subgroup contains the group of inner automorphisms, and is a normal subgroup inside the automorphism group.
Direct product-closedness
This automorphism property is direct product-closed
View a complete list of direct product-closed automorphism properties
Let and be groups and be class automorphisms of respectively. Then, is a class automorphism of .
Here, is the automorphism of that acts as on the first coordinate and on the second.
References
- On the outer automorphisms of a group by William Burnside, Proceedings of the London Mathematical Society, ISSN 1460244X (online), ISSN 00246115 (print), Volume 11, (Year 1913): ^{}^{More info}
- Finite groups with class-preserving outer automorphisms by G. E. Wall, Journal of the London Mathematical Society, ISSN 14697750 (online), ISSN 00246107 (print), Page 315 - 320(Year 1947): ^{}^{More info}
- Class-preserving automorphisms of finite groups by Martin Hertweck, Journal of Algebra, Volume 241, Issue 1, 1 July 2001, Pages 1-26^{More info}
- Class-preserving automorphisms of finite p-groups by Manoj K. Yadav, Journal of the London Mathematical Society, 2007, Page 755-772^{More info}
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