Let $V$ be a vector space over a field $F$ of characteristic $p\geq 0$ and let $T$ be a regular subgroup of the affine group $AGL(V)$. In the finite dimensional case we show that, if $T$ is abelian or $p>0$, then $T$ is unipotent. For $T$ abelian, pushing forward some ideas used in [A. Caranti, F. Dalla Volta and M. Sala, Abelian regular subgroups of the affine group and radical rings, Publ. Math. Debrecen {\bf 69} (2006), 297--308.], we show that the set $\left\{t-I\mid t\in T\right\}$ is a subalgebra of $End_F(F\oplus V)$, which is nilpotent when $V$ has finite dimension. This allows a rather systematic construction of abelian regular subgroups.
A. Caranti, F. Dalla Volta and M. Sala (2006). Abelian regular subgroups of the affine group and radical rings. Publ. Math. Debrecen. 69, 297-308 P. Hegedus (2000). Regular subgroups of the Affine group. J. Algebra. 225, 740-742 J. E. Humphreys (1975). Linear algebraic groups. Springer.
Tamburini Bellani, M. C. (2012). Some remarks on regular subgroups of the affine group. International Journal of Group Theory, 1(1), 17-23. doi: 10.22108/ijgt.2012.468
MLA
Tamburini Bellani, M. C. . "Some remarks on regular subgroups of the affine group", International Journal of Group Theory, 1, 1, 2012, 17-23. doi: 10.22108/ijgt.2012.468
HARVARD
Tamburini Bellani, M. C. (2012). 'Some remarks on regular subgroups of the affine group', International Journal of Group Theory, 1(1), pp. 17-23. doi: 10.22108/ijgt.2012.468
CHICAGO
M. C. Tamburini Bellani, "Some remarks on regular subgroups of the affine group," International Journal of Group Theory, 1 1 (2012): 17-23, doi: 10.22108/ijgt.2012.468
VANCOUVER
Tamburini Bellani, M. C. Some remarks on regular subgroups of the affine group. International Journal of Group Theory, 2012; 1(1): 17-23. doi: 10.22108/ijgt.2012.468
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