3-step group for a prime

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The article defines a property of groups, where the definition may be in terms of a particular prime that serves as parameter
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Definition

Suppose G is a finite group and p is a prime number. We say that G is a 3-step group with respect to p:

  1. O_{p,p'}(G) is a Frobenius group with Frobenius kernel O_p(G) and cyclic complement of odd order. In particular, this means that either p = 2 or G has odd order.
  2. G = O_{p,p',p}(G) and strictly contains O_{p,p'}(G).
  3. G/O_p(G) is a Frobenius group with Frobenius kernel O_{p,p'}(G)/O_p(G).

Examples

Symmetric group:S4 is an example of a 3-step group with p = 2. If G = S_4, then:

References

Textbook references