Counterexamples To a Theorem of Cauchy

• Published in 1968
• Added on 2024-10-09

B. Huppert writes (in [1; p. 304]): "Die folgende, bisher unbewiesene Vermutung stammt schon von Cauchy ([2], S.1199; siehe auch Frobenius [4], S.353): Sei \(p \neq 2\) eine Primzahl, \(\mathfrak{G}\) eine primitive Permutationsgruppe vom Grad \(p+1\). Dann ist \(\mathfrak{G}\) zweifach transitiv." Actually, Cauchy announces without proof a theorem which Frobenius (loc. cit.) and de Séguier ([3; p. 86, note 4]) show to be false, and he deduces from it that primitive groups of degree \(p+1\) (\(p\) an odd prime) are two-fold transitive. Both Frobenius and de Séguier point out that this latter result is nevertheless true for \(p \leq 13\); Huppert proves it with the additional assumption that the groups in question be soluble; and W. R. Scott ([5; §§13.7, 13.8]) gives a verification for groups containing regular subgroups in the cases \(p \leq 37\).

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BibTeX entry

@article{CounterexamplesToaTheoremofCauchy,
	key = {CounterexamplesToaTheoremofCauchy},
	type = {article},
	title = {Counterexamples To a Theorem of Cauchy},
	author = {Peter M. Neumann, Charles C. Sims, James Wiegold},
	abstract = {B. Huppert writes (in [1; p. 304]): "Die folgende, bisher unbewiesene Vermutung stammt schon von Cauchy ([2], S.1199; siehe auch Frobenius [4], S.353): Sei \(p \neq 2\) eine Primzahl, \(\mathfrak{\{}G{\}}\) eine primitive Permutationsgruppe vom Grad \(p+1\). Dann ist \(\mathfrak{\{}G{\}}\) zweifach transitiv." Actually, Cauchy announces without proof a theorem which Frobenius (loc. cit.) and de S{\'{e}}guier ([3; p. 86, note 4]) show to be false, and he deduces from it that primitive groups of degree \(p+1\) (\(p\) an odd prime) are two-fold transitive. Both Frobenius and de S{\'{e}}guier point out that this latter result is nevertheless true for \(p \leq 13\); Huppert proves it with the additional assumption that the groups in question be soluble; and W. R. Scott ([5; §§13.7, 13.8]) gives a verification for groups containing regular subgroups in the cases \(p \leq 37\).},
	comment = {},
	date_added = {2024-10-09},
	date_published = {1968-10-09},
	urls = {https://academic.oup.com/jlms/article-abstract/s1-43/1/234/933632},
	collections = {attention-grabbing-titles,drama},
	url = {https://academic.oup.com/jlms/article-abstract/s1-43/1/234/933632},
	urldate = {2024-10-09},
	year = 1968
}