Projective special linear group is simple

From Groupprops
Revision as of 16:49, 15 May 2015 by Vipul (talk | contribs) (Facts used)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search
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 (?)).

Statement

Let k be a field and n be a natural number greater than 1. Then, the projective special linear group PSL_n(k) is a simple group provided one of these conditions holds:

  • n \ge 3.
  • k has at least four elements.

Facts used

  1. Special linear group is perfect: Under the same conditions (n \ge 3 or k has at least four elements), the special linear group SL_n(k) is a perfect group: it equals its own derived subgroup.
  2. Perfectness is quotient-closed: The quotient of a perfect group by a normal subgroup is perfect.
  3. Abelian normal subgroup of core-free maximal subgroup is contranormal implies derived subgroup of whole group is monolith

Related facts

Related facts about special linear group and projective special linear group

Related facts about simplicity of linear groups

Proof

The proof proceeds in the following steps:

  1. PSL_n(k) satisfies the hypotheses for fact (3): Consider the natural action of PSL_n(k) on the projective space \mathbb{P}^{n-1}(k). 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]
  2. The commutator subgroup of PSL_n(k) is contained in every nontrivial normal subgroup of PSL_n(k): This follows from the previous step and fact (3).
  3. PSL_n(k) equals its own commutator subgroup when n \ge 3 or k has at least four elements: This follows from facts (1) and (2).
  4. PSL_n(k) is simple when n \ge 3 or k has at least four elements: : This follows from the last two steps.