As J.S.Rose noted in his book "A Course on Group Theory" : There is a section of GLn(F) which is isomorphic to PSLn(F), n≥1, F is a field"?. I ask that "What can this section be?"
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1$\begingroup$ What do you mean by section? There is no map from GL_n to PSL_n, only to PGL_n. $\endgroup$– user1688Commented Mar 25, 2011 at 9:56
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1$\begingroup$ @Anton: "In group theory a section of a group G is a group that is (isomorphic to) a quotient group of a subgroup of G." (see en.wikipedia.org/wiki/Section_%28group_theory%29) $\endgroup$– SomeoneCommented Mar 25, 2011 at 11:10
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5$\begingroup$ @Babak: SL_n(F) is a subgroup of GL_n(F), and SL_n(F)/Z(SL_n(F)) is isomorphic to PSL_n(F) (with Z denoting the center). $\endgroup$– SomeoneCommented Mar 25, 2011 at 11:12
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$\begingroup$ In fact, if it is clear, since PSL_n(F) [n/=2, F an infinite field] is a non-abelian simple group, one can see that the GL_n(F) is a insoluble group. Maybe PGL_n is right instead of PSL_n?? $\endgroup$– MikasaCommented Mar 25, 2011 at 12:15
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7$\begingroup$ "Someone" is correct. A section of a group $G$ is group of the form $H/K$, where $H$ is a subgroup of $G$ and $K$ is a normal subgroup of $H$. A section therefore need be neither a subgroup nor a homomorphic image of $G$ itself."Someone" also correctly identifies the necessary subgroup $H$ and normal subgroup $K$ of $H$ in the case $G = {\rm GL}(n,F).$ The statement in Rose's book is accurate, as long as one has a correct understanding of the meaning of the word "section" in this context. I do not see the relevance of the accepted answer below to the original question. $\endgroup$– Geoff RobinsonCommented May 2, 2011 at 7:36
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No clue about the book but there is no non-trivial homomorphism from PSL(2,5) to GL(2,5). Use MAGMA or GAP to check it if it is not clear to you.