Math 2.0

Group theory and generalizations Proposition

Order of an inverse

Builds on Order of an elementLaws of exponents in a group

Statement

Let GG be a group and let g∈Gg \in G. Then gg and its inverse g−1g^{-1} have the same order. In particular, gg has infinite order if and only if g−1g^{-1} has infinite order.

Remarks

For each positive integer nn, (g−1)n=(gn)−1(g^{-1})^n = (g^n)^{-1}, and gn=eg^n = e if and only if (gn)−1=e(g^n)^{-1} = e. For example, in {1,i,−1,−i}\{1, i, -1, -i\}, both ii and i−1=−ii^{-1} = -i have order 44.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

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Group, concept, no evidence levelGroupconceptLaws of exponents in a group, statement, no evidence levelLaws of exponents in …statementOrder of an element, definition, no evidence levelOrder of an elementdefinitionOrder of an inverse, statement, no evidence levelOrder of an inversestatement
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. Laws of exponents in a group statement
  3. Order of an element definition
  4. Order of an inverse statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Order_(group_theory), Order and structure
About this page
Tag0PMT9YJZVW
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Kindstatement, proposition
WritingW0 stub
MSC20A05
StatusThis copy of the site has no build data. Run m2 status 0PMT9YJZVW.
UsesOrder of an element, Laws of exponents in a group

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