Every Sylow subgroup intersects the center nontrivially or is contained in a centralizer

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Statement

Suppose G is a finite group and p is a prime dividing the order of G. Then, every p-Sylow subgroup of G satisfies at least one of these two conditions:

  1. It intersects the center of G nontrivially.
  2. it is contained in the centralizer of a non-central element.

Further, if any one p-Sylow subgroup satisfies a particular condition, so do all the others.

Related facts

Facts used

  1. Class equation of a group
  2. Sylow subgroups exist
  3. Sylow implies order-dominating: Any two Sylow subgroups are conjugate, and any p-subgroup is contained in a p-Sylow subgroup.
  4. Cauchy's theorem for Abelian groups
  5. Central implies normal

Proof

Given: A finite group G of order n = p^rm, where p is prime, r is a positive integer, and p does not divide m.

To prove: Every p-Sylow subgroup of G either intersects the center nontrivially, or is contained in the centralizer of a non-central element.

Proof: Consider the class equation of G (fact (1)):

|G| = |Z(G)| + \sum_{i=1}^r |G:C_G(g_i)|

where c_1,c_2,\dots,c_r are the conjugacy classes of non-central elements and g_i is an element of c_i for each i.

We consider two cases:

  1. Case that p divides the order of Z(G):
    1. There exists a normal subgroup of order p in G: Since Z(G) is Abelian, fact (4) yields that it has a subgroup H of order p. Since H is in the center, H is normal in G (by fact (5)). Thus, H is a normal subgroup of G of order p.
    2. Suppose P is any p-Sylow subgroup of G. By fact (3), the subgroup H is contained in some conjugate of P. Since H is normal, this forces H \le P. Thus, P intersects the center nontrivially -- the intersection contains a subgroup of order p.
  2. Case that p does not divide the order of Z(G):
    1. There exists i such that p does not divide the index k of C_G(g_i) in G: Since p divides the order of G, p cannot divide the index of every C_G(g_i), otherwise the class equation would yield that p divides the order of Z(G).
    2. C_G(g_i) is a proper subgroup of G whose order is a multiple of p^r: Since g_i is non-central, C_G(g_i) is proper in G. Further, since |G:C_G(g_i)| = k is relatively prime to p, Lagrange's theorem (fact (3)) yields that the order of C_G(g_i) is p^rm/k, which is a multiple of p^r.
    3. C_G(g_i) contains a subgroup of order p^r: This follows by fact (2).
    4. Any p-Sylow subgroup of G is of the form gPg^{-1} for some P \le C_G(g_i): This follows from the previous step, and fact (3).
    5. Any p-Sylow subgroup of G is contained in the centralizer of a non-central element: This follows from the previous step; in fact, it is contained in C_G(gg_ig^{-1}).