Transpose-inverse map: Difference between revisions
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===Conditions under which it is and is not an inner automorphism=== | ===Conditions under which it is and is not an inner automorphism=== | ||
* [[Transpose-inverse map is inner on special linear group of degree two]] | * [[Transpose-inverse map is inner automorphism on special linear group of degree two]] | ||
* For the general linear group of degree two over a field, the transpose-inverse map is inner iff the field is [[field:F2]] (this follows from the corresponding observation for class-preserving automorphisms, though it can also be deduced from the fact that the transpose-inverse map is a composite of a radial automorphism -- that is not inner -- with an inner automorphism; see [[endomorphism structure of general linear group of degree two over a finite field]] for details that cross-apply to arbitrary fields). | * For the general linear group of degree two over a field, the transpose-inverse map is inner iff the field is [[field:F2]] (this follows from the corresponding observation for class-preserving automorphisms, though it can also be deduced from the fact that the transpose-inverse map is a composite of a radial automorphism -- that is not inner -- with an inner automorphism; see [[endomorphism structure of general linear group of degree two over a finite field]] for details that cross-apply to arbitrary fields). | ||
* For general and special linear groups of higher degree, the transpose-inverse map is never inner (verification needed). | * For general and special linear groups of higher degree, the transpose-inverse map is never inner (verification needed). | ||
Revision as of 17:50, 12 January 2014
Definition
The transpose-inverse map is an automorphism of the general linear group over any field or ring, and it is defined as the composition of the matrix transpose and the inverse map.
The map has order two.
Facts
- The fixed-point subgroup of this map is termed the orthogonal group.
- The special linear group is invariant (i.e., it is mapped to itself) under this map.
- The transpose-inverse map defines a map (by restriction) on the special linear group and also induces maps on the projective general linear group and projective special linear group.
Conditions under which it is and is not an inner automorphism
- Transpose-inverse map is inner automorphism on special linear group of degree two
- For the general linear group of degree two over a field, the transpose-inverse map is inner iff the field is field:F2 (this follows from the corresponding observation for class-preserving automorphisms, though it can also be deduced from the fact that the transpose-inverse map is a composite of a radial automorphism -- that is not inner -- with an inner automorphism; see endomorphism structure of general linear group of degree two over a finite field for details that cross-apply to arbitrary fields).
- For general and special linear groups of higher degree, the transpose-inverse map is never inner (verification needed).
Conditions under which it is and is not a class-preserving automorphism
Conditions under which it is and is not a class-inverting automorphism
- Transpose-inverse map is class-inverting automorphism for general linear group
- Transpose-inverse map induces class-inverting automorphism on projective general linear group
- The transpose-inverse map does not induce a class-inverting automorphism on the special linear group (unless the field size is two). However, special linear group of degree two has a class-inverting automorphism (different from the transpose-inverse map).