3-step group for a prime

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. $O_{p,p^{\prime }}(G)$ 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. $G=O_{p,p^{\prime },p}(G)$ and strictly contains $O_{p,p^{\prime }}(G)$.
3. $G/O_p(G)$ is a Frobenius group with Frobenius kernel $O_{p,p^{\prime }}(G)/O_p(G)$.

Examples

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

• $O_p(G)$ is the normal Klein four-subgroup of symmetric group:S4, isomorphic to the Klein four-group.
• $O_{p,p^{\prime }}(G)$ is A4 in S4, isomorphic to alternating group:A4. This is a Frobenius group with $O_p(G)$ as its Frobenius kernel and complement cyclic group:Z3.
• $G/O_p(G)$ is isomorphic to symmetric group:S3, and has Frobenius kernel A3 in S3, which is the image mod $O_p(G)$ of A4 in S4, which is $O_{p,p^{\prime }}(G)$.

References

Textbook references

• Finite Groups by Daniel Gorenstein, ISBN 0821843427, Page 401, Section 14.1 (Basic Properties of CN-Groups), ^{More info}