Difference between revisions of "Sylow Theory"
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Kfagerstrom (talk | contribs) (Created page with " ====Sylow Theorem 1 (Existence) ==== If <math>p </math> is prime and <math> p^k|n </math>, then there exists <math> H<G </math> with <math>|H|=p^k</math>") |
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| − | + | Let G be a finite group and p a prime. Write the order of <math> G</math> as | |
| − | + | <math> |G| = p^em </math> where <math>p \not| m</math>. A Sylow p-subgroup of <math> G</math> is a subgroup <math> H \leq G </math>such that <math> |H| = p^e</math>. That is, a Sylow p-subgroup of <math>G </math> is a subgroup whose order is the highest conceivable power of <math>p </math> according to Lagrange’s Theorem. | |
| + | We set <math> \text{Syl}_p(G)</math> to be the collection of all Sylow p-subgroups of <math>G </math> and <math>n_p = | \text{Syl}_p(G)| </math> to be the number of Sylow p-subgroups. | ||
Revision as of 15:50, 16 January 2023
Let G be a finite group and p a prime. Write the order of \( G\) as
\( |G| = p^em \) where \(p \not| m\). A Sylow p-subgroup of \( G\) is a subgroup \( H \leq G \)such that \( |H| = p^e\). That is, a Sylow p-subgroup of \(G \) is a subgroup whose order is the highest conceivable power of \(p \) according to Lagrange’s Theorem.
We set \( \text{Syl}_p(G)\) to be the collection of all Sylow p-subgroups of \(G \) and \(n_p = | \text{Syl}_p(G)| \) to be the number of Sylow p-subgroups.
