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.
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
Dixon, M., Evans, M., & Smith, H. (2013). Omissible extensions of SL2(k) where k is a field of positive characteristic. International Journal of Group Theory, 2(1), 145-155. doi: 10.22108/ijgt.2013.2739
MLA
Martyn Dixon; Martin Evans; Howard Smith. "Omissible extensions of SL2(k) where k is a field of positive characteristic". International Journal of Group Theory, 2, 1, 2013, 145-155. doi: 10.22108/ijgt.2013.2739
HARVARD
Dixon, M., Evans, M., Smith, H. (2013). 'Omissible extensions of SL2(k) where k is a field of positive characteristic', International Journal of Group Theory, 2(1), pp. 145-155. doi: 10.22108/ijgt.2013.2739
VANCOUVER
Dixon, M., Evans, M., Smith, H. Omissible extensions of SL2(k) where k is a field of positive characteristic. International Journal of Group Theory, 2013; 2(1): 145-155. doi: 10.22108/ijgt.2013.2739
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