Math 2.0

Group theory and generalizations Definition

Normal core

Builds on Conjugate subgroupNormal subgroupIntersection of subgroups is a subgroup

Definition

Let GG be a group and let HH be a subgroup of GG. The normal core of HH in GG is the subgroup Core⁡G(H)=⋂g∈GgHg−1\operatorname{Core}_G(H) = \bigcap_{g \in G} gHg^{-1}.

Remarks

The core is a normal subgroup of GG, and it contains each normal subgroup of GG that lies in HH. The subgroup HH is normal exactly when Core⁡G(H)=H\operatorname{Core}_G(H) = H. The core is the kernel of the action of GG on the left cosets of HH by left multiplication.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

Show the path as a graph
Group, concept, no evidence levelGroupconceptConjugate elements, definition, no evidence levelConjugate elementsdefinitionSubgroup, definition, no evidence levelSubgroupdefinitionConjugate subgroup, definition, no evidence levelConjugate subgroupdefinitionSubgroup criterion, statement, no evidence levelSubgroup criterionstatementNormal subgroup, definition, no evidence levelNormal subgroupdefinitionIntersection of subgroups is a subgroup, statement, no evidence levelIntersection of subgr…statementNormal core, definition, no evidence levelNormal coredefinition
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.
  • no evidence level
  1. Group concept
  2. Conjugate elements definition
  3. Subgroup definition
  4. Conjugate subgroup definition
  5. Subgroup criterion statement
  6. Normal subgroup definition
  7. Intersection of subgroups is a subgroup statement
  8. Normal core definition

Where this leads

25 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Core_(group_theory), Definition
About this page
Tag0T0KMZPYV9
URIm2://0T0KMZPYV9
Kinddefinition
WritingW0 stub
MSC20D25
DefinesNormal core
UsesConjugate subgroup, Normal subgroup, Intersection of subgroups is a subgroup

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by