Projective special linear group is simple
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This article gives the statement, and possibly proof, of a particular group or type of group (namely, Projective special linear group (?)) satisfying a particular group property (namely, Simple group (?)).
- has at least four elements.
- Special linear group is perfect: Under the same conditions ( or has at least four elements), the special linear group is a perfect group: it equals its own derived subgroup.
- Perfectness is quotient-closed: The quotient of a perfect group by a normal subgroup is perfect.
- Abelian normal subgroup of core-free maximal subgroup is contranormal implies derived subgroup of whole group is monolith
Related facts about special linear group and projective special linear group
- Special linear group is perfect
- Special linear group is quasisimple
- Projective special linear group equals alternating group in only finitely many cases
Related facts about simplicity of linear groups
- Projective symplectic group is simple
- Projective special orthogonal group for bilinear form of positive Witt index is simple
- projective special orthogonal group over reals is simple
The proof proceeds in the following steps:
- satisfies the hypotheses for fact (3): Consider the natural action of on the projective space . This is a primitive group action, and the stabilizer of any point is thus a core-free maximal subgroup. PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
- The commutator subgroup of is contained in every nontrivial normal subgroup of : This follows from the previous step and fact (3).
- equals its own commutator subgroup when or has at least four elements: This follows from facts (1) and (2).
- is simple when or has at least four elements: : This follows from the last two steps.