Finite group having at least two conjugacy classes of involutions has order less than the cube of the maximum of orders of centralizers of involutions

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Statement

Suppose G is a finite group such that there are at least two conjugacy classes of involutions (non-identity elements of order two) in G. Then, if h is the maximum of the orders of all subgroups of G that arise as a Centralizer of involution (?), we have:

|G| < h^3.

Related facts

Facts used

  1. Involutions are either conjugate or have an involution centralizing both of them
  2. Group acts as automorphisms by conjugation
  3. Fundamental theorem of group actions

Proof

Given: A group G with at least two conjugacy classes of involutions. h is the maximum possible order of a subgroup arising as the centralizer of an involution of G.

To prove: |G| < h^3.

Proof:

  1. Let x be an involution of G such that |C_G(x)| has order h and let C = C_G(x). Note that x exists by the definition of h.
  2. Let y be an involution of G that is not conjugate to x and let D = C_g(y): Note that y exists because of the assumption that there are at least two conjugacy classes of involutions.
  3. Let y = y_1, y_2, \dots, y_t be distinct involutions of D, and define D_i = C_G(y_i). In particular, D_1 = D.
  4. t \le |D| \le h: t \le |D| because the y_i are all distinct elements of D. |D| \le h because D is a centralizer of involution and by definition, h is the maximum of the orders of such subgroups.
  5. The number of distinct non-identity elements in \bigcup_{i=1}^t D_i is less than h^2: Each D_i has at most h - 1 non-identity elements, and there are t of them, with t \le h. So, the total number of elements is less than h^2.
  6. Suppose x has a total of m conjugates x_1, x_2, \dots, x_m. Then, each x_j is contained in the union \bigcup_{i=1}^t D_i: Any x_j is conjugate to x. Since by assumption x is not conjugate to y, x_j is not conjugate to y. Thus, by fact (1), there exists an involution centralizing both x_j and y. This involution lives in C_G(y) so it is some y_i. Thus, x_j \in C_G(y_i) for some i.
  7. m < h^2: This follows from the previous two steps.
  8. |G| < h^3: First, note that under the action of the group on itself by conjugation, the centralizer of x is its stabilizer, so by fact (3), the coset space of C_G(x) in G is in bijection with the conjugacy class of x. Thus, m = [G:C_G(x)]. By fact (4) (Lagrange's theorem), we get m = |G|/h. Since m < h^2 by the previous step, we get |G| < h^3.

References

Textbook references