Math 2.0

statement proposition

Subgroups of abelian groups are normal

Statement

Let GG be an abelian group and let HH be a subgroup of GG. Then HH is normal in GG.

Remarks

For g∈Gg \in G and h∈Hh \in H, commutativity gives ghg−1=hgg−1=h∈Hghg^{-1} = hgg^{-1} = h \in H. The converse is false: in the quaternion group Q8Q_8 each subgroup is normal, but Q8Q_8 is not abelian. More generally, each subgroup of the center of any group is normal.

Explanations

L1 human

In an abelian group, conjugation does nothing, because ghg−1=hghg^{-1} = h. So a subgroup can never be moved by conjugation. Thus all subgroups are normal.

What this node 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 node is. They do not measure its truth.

What this needs

To read this node, read these 9 nodes in this order. The foundations come first, and the list ends with this node.

Binary operation, definition, no evidence levelBinary operationdefinitionGroup, concept, no evidence levelGroupconceptCommutativity, definition, no evidence levelCommutativitydefinitionConjugate elements, definition, no evidence levelConjugate elementsdefinitionSubgroup, definition, no evidence levelSubgroupdefinitionAbelian group, definition, no evidence levelAbelian groupdefinitionConjugate subgroup, definition, no evidence levelConjugate subgroupdefinitionNormal subgroup, definition, no evidence levelNormal subgroupdefinitionSubgroups of abelian groups are normal, statement, no evidence levelSubgroups of abelian …statement
Each node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
  • no evidence level
  1. Binary operation definition
  2. Group concept
  3. Commutativity definition
  4. Conjugate elements definition
  5. Subgroup definition
  6. Abelian group definition
  7. Conjugate subgroup definition
  8. Normal subgroup definition
  9. Subgroups of abelian groups are normal statement

Sources

  1. Web https://en.wikipedia.org/wiki/Normal_subgroup, Examples

What links here

No node links to this node yet.