Math 2.0

concept

Generators of a group

Summary

A subset SS of a group GG generates GG when the subgroup generated by SS is all of GG. Then each element of GG is a finite product of elements of SS and of their inverses. The elements of SS are called generators of GG.

Remarks

The set {1}\{1\} generates (Z,+)(\mathbb{Z}, +), and so does the set {2,3}\{2, 3\}. Each group is generated by the set of all its elements. A group generated by one element is a cyclic group, and a group generated by a finite set is a finitely generated group.

Explanations

L1 human

Generators are building blocks for a group: you can reach each element from them with products and inverses. A small set of generators gives a short description of a big group. For example, all symmetries of a square come from one rotation and one reflection.

What this node needs for W2
  • edge defines in: at least one definition (0 of 1)
  • edge instance_of in: at least two examples (0 of 2)
  • edge not_instance_of in: at least one non-example (0 of 1)
  • edge specializes both: the more general notions and the special cases (0 of 1)

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What this needs

To read this node, read these 6 nodes in this order. The foundations come first, and the list ends with this node.

Group, concept, no evidence levelGroupconceptSubgroup, definition, no evidence levelSubgroupdefinitionSubgroup criterion, statement, no evidence levelSubgroup criterionstatementIntersection of subgroups is a subgroup, statement, no evidence levelIntersection of subgr…statementSubgroup generated by a subset, definition, no evidence levelSubgroup generated by…definitionGenerators of a group, concept, no evidence levelGenerators of a groupconcept
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. Group concept
  2. Subgroup definition
  3. Subgroup criterion statement
  4. Intersection of subgroups is a subgroup statement
  5. Subgroup generated by a subset definition
  6. Generators of a group concept

Sources

  1. Web https://en.wikipedia.org/wiki/Generating_set_of_a_group, lead section
  2. Web https://en.wikipedia.org/wiki/Generating_set_of_a_group, Examples

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