3-step group for a prime
From Groupprops
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 is a finite group and is a prime number. We say that is a 3-step group with respect to :
- is a Frobenius group with Frobenius kernel and cyclic complement of odd order. In particular, this means that either or has odd order.
- and strictly contains .
- is a Frobenius group with Frobenius kernel .
Examples
Symmetric group:S4 is an example of a 3-step group with . If , then:
- is the normal Klein four-subgroup of symmetric group:S4, isomorphic to the Klein four-group.
- is A4 in S4, isomorphic to alternating group:A4. This is a Frobenius group with as its Frobenius kernel and complement cyclic group:Z3.
- is isomorphic to symmetric group:S3, and has Frobenius kernel A3 in S3, which is the image mod of A4 in S4, which is .
References
Textbook references
- Finite Groups by Daniel Gorenstein, ISBN 0821843427, Page 401, Section 14.1 (Basic Properties of CN-Groups), ^{More info}