Cauchy's theorem (group theory) | Wikipedia audio article
Channel: wikipedia tts
Date: 28/12/2018

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Cauchy%27s_theorem_(group_theory)


00:01:24 1 Statement and proof
00:02:06 1.1 Proof 1
00:02:24 1.2 Proof 2
00:03:24 2 Uses
00:04:46 3 References
00:06:44 4 External links
00:08:14 e. Counting the elements of X by orbits, and reducing modulo p, one sees that the number of elements satisfying xp
00:08:38 e is one such element, so there must be at least p − 1 other solutions for x, and these solutions are elements of order p. This completes the proof.
00:08:54 Uses



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SUMMARY
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Cauchy's theorem is a theorem in the mathematics of group theory, named after Augustin Louis Cauchy. It states that if G is a finite group and p is a prime number dividing the order of G (the number of elements in G), then G contains an element of order p. That is, there is x in G so that p is the lowest non-zero number with xp = e, where e is the identity element.
The theorem is related to Lagrange's theorem, which states that the order of any subgroup of a finite group G divides the order of G. Cauchy's theorem implies that for any prime divisor p of the order of G, there is a subgroup of G whose order is p—the cyclic group generated by the element in Cauchy's theorem.
Cauchy's theorem is generalised by Sylow's first theorem, which implies that if pn is the maximal power of p dividing the order of G, then G has a subgroup of order pn (and using the fact that a p-group is solvable, one can show that G has subgroups of order pr for any r less than or equal to n).