Alternative Definitions
There are a number of group theoretical properties which are interesting on their own right, but which happen to be equivalent to the group possessing a permutation representation that makes it a Frobenius group.
- G is a Frobenius group if and only if G has a proper, nonidentity subgroup H such that H ∩ Hg is the identity subgroup for every g ∈ G − H, i.e. H is a malnormal subgroup of G.
This definition is then generalized to the study of trivial intersection sets which allowed the results on Frobenius groups used in the classification of CA groups to be extended to the results on CN groups and finally the odd order theorem.
Assuming that is the semidirect product of the normal subgroup K and complement H, then the following restrictions on centralizers are equivalent to G being a Frobenius group with Frobenius complement H:
- The centralizer CG(k) is a subgroup of K for every nonidentity k in K.
- CH(k) = 1 for every nonidentity k in K.
- CG(h) ≤ H for every nonidentity h in H.
Read more about this topic: Frobenius Group
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