Difference between revisions of "Sylow Theory"
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Let G be a finite group and p a prime. Write the order of <math> G</math> as | 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. | <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. | 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 20:38, 17 January 2023
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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.
