Math 2.0

Group theory and generalizations Example

The special linear group is the kernel of the determinant

Builds on General linear groupSpecial linear groupDeterminantKernel of a homomorphism

Example

Let n≥1n \ge 1. The determinant is a homomorphism det⁡:GLn(ℝ)→ℝ×\det : \mathrm{GL}_n(\mathbb{R}) \to \mathbb{R}^\times, because det⁡(AB)=det⁡(A)det⁡(B)\det(AB) = \det(A)\det(B). Here GLn(ℝ)\mathrm{GL}_n(\mathbb{R}) is the general linear group, and ℝ×\mathbb{R}^\times is the group of nonzero real numbers under multiplication. The kernel of det⁡\det is the set of matrices with determinant 11. This is the special linear group SLn(ℝ)\mathrm{SL}_n(\mathbb{R}). Thus SLn(ℝ)\mathrm{SL}_n(\mathbb{R}) is a normal subgroup of GLn(ℝ)\mathrm{GL}_n(\mathbb{R}).

Remarks

For n≥2n \ge 2, the group GLn(ℝ)\mathrm{GL}_n(\mathbb{R}) is not abelian. Thus the normality does not come from commutativity.

Reading path

To understand this page from the ground up, read these 14 pages in order. The foundations come first.

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General linear group, object, no evidence levelGeneral linear groupobjectGroup, concept, no evidence levelGroupconceptIdentity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionVector space, definition, no evidence levelVector spacedefinitionGroup homomorphism, definition, no evidence levelGroup homomorphismdefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionLinear map, definition, no evidence levelLinear mapdefinitionKernel of a homomorphism, definition, no evidence levelKernel of a homomorph…definitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionDeterminant, definition, no evidence levelDeterminantdefinitionSpecial linear group, object, no evidence levelSpecial linear groupobjectThe special linear group is the kernel of the determinant, example, no evidence levelThe special linear gr…example
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. General linear group object
  2. Group concept
  3. Identity matrix object
  4. Matrix definition
  5. Vector space definition
  6. Group homomorphism definition
  7. Square matrix definition
  8. Linear map definition
  9. Kernel of a homomorphism definition
  10. Multilinear map definition
  11. Alternating multilinear map definition
  12. Determinant definition
  13. Special linear group object
  14. The special linear group is the kernel of the determinant example

Further reading

  1. Web https://en.wikipedia.org/wiki/Special_linear_group, lead section
About this page
Tag556HZEYCQF
URIm2://556HZEYCQF
Kindexample
WritingW0 stub
MSC20G15
Instance ofKernel of a homomorphism, Normal subgroup
IllustratesThe kernel is a normal subgroup
UsesGeneral linear group, Special linear group, Determinant, Kernel of a homomorphism

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