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Sylow p-Subgroups of A 5

Sylow p-Subgroups of A 5. Christopher Kroll. Sylow’s First Theorem:. Let G be a finite group and let p be a prime. If p k divides | G| then G has at least one subgroup of order p k .

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Sylow p-Subgroups of A 5

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  1. Sylow p-Subgroups of A5 Christopher Kroll

  2. Sylow’s First Theorem: • Let G be a finite group and let p be a prime. If pkdivides |G| then G has at least one subgroup of order pk. • Example: Since |A5| = 60 = (22)(3)(5), Sylow’s First Theorem confirms the existence of subgroups of order 2, 22, 3 and 5.

  3. Sylow p-Subgroups • Let G be a finite group and let p be a prime divisor of |G|. If pkdivides |G| and pk+1 does not divide |G|, then any subgroup of G of order pkis called a Sylow p-Subgroup of G. • Example: In A5 the Sylow 2-Subgroup will have order 22.

  4. The Sylow 5-Subgroups of A5 • A5 contains: • 15 Sylow 2-Subgroups ( These are isomorphic to Z2 + Z2 , a.k.a the Klein 4 group) • 10 Sylow 3-Subgroups. • 6 Sylow 5-Subgroups. • These subgroups can be demonstrated visually with the icosahedron.

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