Sylow theorems | Wikipedia audio article
Channel: wikipedia tts
Date: 2018-12-28

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Sylow_theorems


00:01:34 1 Theorems
00:01:59 1.1 Consequences
00:03:49 1.2 Sylow theorems for infinite groups
00:03:58 2 Examples
00:04:49 3 Example applications
00:05:22 3.1 Cyclic group orders
00:05:36 3.2 Small groups are not simple
00:05:45 3.3 Wilson's theorem
00:06:54 3.4 Fusion results
00:07:45 4 Proof of the Sylow theorems
00:08:04 5 Algorithms
00:08:50 6 See also
00:08:57 7 Notes
00:09:07 8 References
00:10:16 8.1 Proofs
00:10:51 8.2 Algorithms
00:11:08 9 External links
00:11:42 6, since n5 must divide 6 (
00:12:10 42
00:12:23 2 · 3) and n7 must equal 1 (mod 7), so n7
00:12:43 60
00:12:55 10 and n5
00:13:21 Wilson's theorem
00:14:13 Fusion results
00:15:09 Bh−1
00:15:39 PK and P∩K
00:16:30 Proof of the Sylow theorems
00:17:31 pkm
00:18:07 {g ∈ G | gω
00:18:30 hα implies g
00:20:56 νp(m)
00:21:36 νp(|Gω|) ≤ νp(|Ω|)
00:21:50 psv where p does not divide v. By the orbit-stabilizer theorem we have |Gω|
00:24:03 gP so g−1HgP
00:24:21 |P|
00:25:01 [G : StabG(P)]. StabG(P)
00:25:09 P }
00:25:23 |G : NG(P)|, and it follows that this number is a divisor of |G|/q. Let Ω be the set of all Sylow p-subgroups of G, and let P act on Ω by conjugation. Let Q ∈ Ω0 and observe that then Q
00:26:02 Q.  It follows that Ω0
00:26:09 1 (mod p).
00:26:13 Algorithms
00:27:45 See also



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SUMMARY
=======
In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow (1872) that give detailed information about the number of subgroups of fixed order that a given finite group contains. The Sylow theorems form a fundamental part of finite group theory and have very important applications in the classification of finite simple groups.
For a prime number p, a Sylow p-subgroup (sometimes p-Sylow subgroup) of a group G is a maximal p-subgroup of G, i.e., a subgroup of G that is a p-group (so that the order of every group element is a power of p) that is not a proper subgroup of any other p-subgroup of G. The set of all Sylow p-subgroups for a given prime p is sometimes written Sylp(G).
The Sylow theorems assert a partial converse to Lagrange's theorem. Lagrange's theorem states that for any finite group G the order (number of elements) of every subgroup of G divides the order of G.  The Sylow theorems state that for every prime factor p of the order of a finite group G, there exists a Sylow p-subgroup of G of order pn, the highest power of p that divides the order of G.  Moreover, every subgroup of order pn is a Sylow p-subgroup of G, and the Sylow p-subgroups of a group (for a given prime p) are conjugate to each other. Furthermore, the number of Sylow p-subgroups of a group for a given prime p is congruent to 1 mod p.