# Omissible extensions of $SL_2(k)$ where $k$ is a field of positive characteristic

Document Type : Ischia Group Theory 2012

Authors

1 The University of Alabama

2 Bucknell University

Abstract

A normal subgroup $N$ of a group $G$ is said to be an‎ omissible subgroup of $G$ if it has the following property‎: ‎whenever $X\leq G$ is such that $G=XN$‎, ‎then $G=X$‎. ‎In this note we construct various groups $G$‎, ‎each of which has an omissible subgroup $N\neq 1$ such that $G/N\cong SL_2(k)$ where $k$ is a field of positive characteristic‎.

Keywords

Main Subjects

#### References

J. V. Brawley and G. E. Schnibben (1989). Infinite Agebraic Extensions of Finite Fields. Contemporary Mathematics, American Math. Soc., Providence, RI.. 95 Contemporary Mathematics, American Math. Soc., Providence, RI. (2000). Topics in Geometric Group Theory. Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL. M. R. Dixon and M. J. Evans (1999). Groups with the minimum condition on insoluble subgroups. Arch. Math. (Basel). 72, 241-251 M. R. Dixon, M. J. Evans and H. Smith (2005). Groups with all proper subgroups soluble-by-finite rank. J. Algebra. 289, 135-147 S. D. Kozlov (1991). Frattini extensions of the projective special linear group. Sibirsk. Math. Zh., 32 no. 2 (1991) 88--93. English translation in: Siberian Math. J.. 32 (2), 252-256 H. Matsumura (1989). Commutative Ring Theory. Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge. 8 D. J. S. Robinson (1996). A course in the theory of groups. Graduate Texts in Mathematics, Springer Verlag, Berlin, Heidelberg, New York. 80 J-P. Serre (1968). Abelian $\ell$-adic Representations and Elliptic Curves. W. A. Benjamin, Inc., New York-Amsterdam. J-P. Serre (1979). Local Fields. Graduate Texts in Mathematics, Springer-Verlag, New York, Berlin. 67