Burnside's lemma

lemma stating that, given a finite group G acting on a set, the number of orbits times |G| equals the sum (over every element of G) of the numbers of fixed points
Intangible theorem Q1330377
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Burnside's lemma

Summary

Burnside's lemma is a theorem[1]. It ranks in the top 8% of theorem entities by monthly Wikipedia readership (219 views/month).[2]

Key Facts

  • Burnside's lemma is credited with the discovery of Augustin-Louis Cauchy[3].
  • Burnside's lemma is credited with the discovery of Ferdinand Georg Frobenius[4].
  • Burnside's lemma's instance of is recorded as theorem[5].
  • William Burnside is named after Burnside's lemma[6].
  • Augustin-Louis Cauchy is named after Burnside's lemma[7].
  • Ferdinand Georg Frobenius is named after Burnside's lemma[8].
  • Burnside's lemma's maintained by WikiProject is recorded as WikiProject Mathematics[9].
  • Burnside's lemma's copyright status is recorded as public domain[10].

Body

Definition and Type

Burnside's lemma's instance of is recorded as theorem[5].

Origins

Things named after include William Burnside[6], a mathematician[11], 1852–1927[12], of United Kingdom[13], awarded the Fellow of the Royal Society[14], specialised in group theory[15]; Augustin-Louis Cauchy[7], a mathematician[16], 1789–1857[17], of France[18], awarded the Pour le Mérite for Sciences and Arts order[19], specialised in mathematical analysis[20]; and Ferdinand Georg Frobenius[8], a mathematician[21], 1849–1917[22], of Kingdom of Prussia[23], specialised in algebra[24].

Why It Matters

Burnside's lemma ranks in the top 8% of theorem entities by monthly Wikipedia readership (219 views/month).[2] It has Wikipedia articles in 16 language editions, a strong signal of global cultural recognition.[25] It is known by 17 alternative names across languages and contexts.[26]

References

Programmatic citations — every numbered marker resolves to a verifiable graph row below.

Direct Wikidata claims

  1. [5] . wikidata.org.
  2. [3] . wikidata.org.
  3. [4] . wikidata.org.
  4. [6] . wikidata.org.
  5. [7] . wikidata.org.
  6. [8] . wikidata.org.
  7. [9] . wikidata.org.
  8. [10] . wikidata.org.

Inline context (facts about related entities)

  1. [11] . Wikidata. wikidata.org. → on this site
  2. [12] . Wikidata. wikidata.org. → on this site
  3. [13] . Wikidata. wikidata.org. → on this site
  4. [14] . Wikidata. wikidata.org. → on this site
  5. [15] . Wikidata. wikidata.org. → on this site
  6. [16] . Wikidata. wikidata.org. → on this site
  7. [17] . Wikidata. wikidata.org. → on this site
  8. [18] . Wikidata. wikidata.org. → on this site
  9. [19] . Wikidata. wikidata.org. → on this site
  10. [20] . Wikidata. wikidata.org. → on this site
  11. [21] . Wikidata. wikidata.org. → on this site
  12. [22] . Wikidata. wikidata.org. → on this site
  13. [23] . Wikidata. wikidata.org. → on this site
  14. [24] . Wikidata. wikidata.org. → on this site

Class ancestry

  1. [1] . Wikidata. wikidata.org.

Aggregate / graph-position facts

  1. [2] . Wikimedia Foundation. dumps.wikimedia.org.
  2. [25] . Wikidata sitelinks. wikidata.org.
  3. [26] . Wikidata aliases. wikidata.org.

📑 Cite this page

Use these citations when quoting this entity in research, articles, AI prompts, or wherever provenance matters. We aggregate Wikidata + Wikipedia + authoritative open-data sources; the stitched, scored, cross-referenced view is what 4ort.xyz contributes.

APA 4ort.xyz Knowledge Graph. (2026). Burnside's lemma. Retrieved May 3, 2026, from https://4ort.xyz/entity/burnside-s-lemma
MLA “Burnside's lemma.” 4ort.xyz Knowledge Graph, 4ort.xyz, 3 May. 2026, https://4ort.xyz/entity/burnside-s-lemma.
BibTeX @misc{4ortxyz_burnside-s-lemma_2026, author = {{4ort.xyz Knowledge Graph}}, title = {{Burnside's lemma}}, year = {2026}, url = {https://4ort.xyz/entity/burnside-s-lemma}, note = {Accessed: 2026-05-03}}
LLM prompt According to 4ort.xyz Knowledge Graph (aggregator of Wikidata, Wikipedia, and authoritative open-data sources): Burnside's lemma — https://4ort.xyz/entity/burnside-s-lemma (retrieved 2026-05-03)

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Edit History

Rolling log of changes to this entity's Wikidata record. Values shown reflect the current state of each edited property — follow the history link to see the precise diff for any edit.

  1. 15d ago · Mynus grey · 2026-07-02 view diff on Wikidata ↗
    Named after
    Instance of theorem
    P14534 MulAction.sum_card_fixedBy_eq_card_orbits_mul_card_group
    Instance of
    + 5 other properties edited (see Wikidata diff for full list)
    "/* wbcreateclaim-create:1| */ [[Property:P14534]]: MulAction.sum_card_fixedBy_eq_card_orbits_mul_card_group, #quickstatements; #temporary_batch_1783013068346"
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