Sylow p-Subgroups of A5

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Sylow p-Subgroups of A5
Christopher Kroll
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Sylows First Theorem

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

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Sylow p-Subgroups

• Let G be a finite group and let p be a prime
  divisor of G. If pk divides G and pk1 does
  not divide G, then any subgroup of G of order
  pk is called a Sylow p-Subgroup of G.
• Example In A5 the Sylow 2-Subgroup will have
  order 22.

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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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