Sylow Theorems

1
Sylow Theorems

• See-low

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Conjugates

• If there exists x in G, G a group, such that
  xa(x-1)b (a, b in G), then a and b are
  conjugates
• Conjugacy is an equivalence relation on G

3
Conjugates

• Cl(a) xa(x-1) x in G, G a group, Is the
  conjugacy class of a
• Distinct conjugacy classes partition a group
• cl(a) GC(a)
• If G is finite, cl(a) divides G

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P-Groups

• If G pn, p prime, n gt 0, G is a p-group
• If G is a p-group, Z(G) gt 1
• Divisibility of Gs order and of each subgroup
• Groups of order p2 are abelian
• Previous result and Lagrange

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Sylow 1

• Existence of subgroups of prime-power order
• G finite, p prime, pk G -gt G has at least
  one subgroup of order pk
• Form factor groups, use Lagrange

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Sylow p-subgroup

• If p is prime, and pk is the largest power of p
  that divides G, then a subgroup of order pk is
  called a p-subgroup of G
• If 2k G and 2(k1) / G, then a subgroup
  of order 2k is called a 2-subgroup of G

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Sylow 2

• If H is a subgroup of G and H pk for some k
  in N then H is contained in some Sylow p-subgroup
  of G.
• Automorphism, homomorphism, orbit, divisibility

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Conjugate Subgroups

• H, K, subgroups of G, are conjugate in G if there
  exists a g in G s.t. H gK(g-1)

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Sylow 3

• The number of sylow p-subgroups of G is 1 (mod p)
  and divides G also, any two Sylow p-subgroups
  are conjugate

10
Cyclic groups of order pq

• If G pq (p, q prime, WLOG pltq), and p /
  (q-1), then G is cyclic (and isomorphic to Zpq

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Problems that Sylow theorems make it easy (or
possible) to solve

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