Math 2.0

Group theory and generalizations Proposition

Tower law for indices

Builds on Index of a subgroup

Statement

Let GG be a group, let HH be a subgroup of GG, and let KK be a subgroup of HH. Then [G:K]=[G:H]⋅[H:K][G : K] = [G : H] \cdot [H : K], where [⋅:⋅][\cdot : \cdot] is the index.

Remarks

The equation holds as an equation of cardinal numbers; in particular [G:K][G : K] is finite if and only if [G:H][G : H] and [H:K][H : K] are both finite. For example, in (ℤ,+)(\mathbb{Z}, +) with H=2ℤH = 2\mathbb{Z} and K=6ℤK = 6\mathbb{Z}, the indices are [ℤ:6ℤ]=6=2⋅3[\mathbb{Z} : 6\mathbb{Z}] = 6 = 2 \cdot 3.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

Show the path as a graph
Group, concept, no evidence levelGroupconceptSubgroup, definition, no evidence levelSubgroupdefinitionLeft coset, definition, no evidence levelLeft cosetdefinitionIndex of a subgroup, definition, no evidence levelIndex of a subgroupdefinitionTower law for indices, statement, no evidence levelTower law for indicesstatement
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. Subgroup definition
  3. Left coset definition
  4. Index of a subgroup definition
  5. Tower law for indices statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Index_of_a_subgroup, Properties
About this page
Tag0ZQJ7KEFQ5
URIm2://0ZQJ7KEFQ5
Aliasesmultiplicativity-of-the-index
Kindstatement, proposition
WritingW0 stub
MSC20A05
StatusThis copy of the site has no build data. Run m2 status 0ZQJ7KEFQ5.
UsesIndex of 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

No page links to this page yet.