On the automorphism group of a linear algebraic monoid

Author:
Mohan S. Putcha

Journal:
Proc. Amer. Math. Soc. **88** (1983), 224-226

MSC:
Primary 20M10; Secondary 20G99

DOI:
https://doi.org/10.1090/S0002-9939-1983-0695247-3

MathSciNet review:
695247

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Abstract: Let be a connected regular monoid with zero. It is shown that an automorphism of is inner if and only if it sends each idempotent of to a conjugate idempotent. In the language of semigroup theory, the automorphism group of maps homomorphically into the automorphism group of the finite lattice of -classes of , and the kernel of this homomorphism is the group of inner automorphisms of . In particular, if the -classes of are linearly ordered, then every automorphism of is inner.

**[1]**J. E. Humphreys,*Linear algebraic groups*, Springer-Verlag, Berlin and New York, 1981. MR**0396773 (53:633)****[2]**M. S. Putcha,*Linear algebraic semigroups*, Semigroup Forum**22**(1981), 287-309. MR**619186 (82g:20108)****[3]**-,*Connected algebraic monoids*, Trans. Amer. Math. Soc.**272**(1982), 693-709. MR**662061 (84d:20066)****[4]**-,*A semigroup approach to linear algebraic groups*, J. Algebra (to appear). MR**690712 (84j:20045)****[5]**-,*Reductive groups and regular semigroups*, J. Algebra (submitted).**[6]**-,*Idempotent cross-sections of**-classes*, Semigroup Forum (to appear).**[7]**L. Renner,*Algebraic monoids*, Ph. D. Thesis, Univ. of British Columbia, 1982.**[8]**-,*Reductive monoids are von-Neumann regular*(to appear).**[9]**T. A. Springer,*Linear algebraic groups*, Birkhaüser, Basel, 1981. MR**632835 (84i:20002)**

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DOI:
https://doi.org/10.1090/S0002-9939-1983-0695247-3

Keywords:
Monoid,
algebraic,
automorphism,
-class

Article copyright:
© Copyright 1983
American Mathematical Society