# Examples of Commutative Rings by Harry C. Hutchins By Harry C. Hutchins

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Example text

13. Primitive Groups with Transitive Subgroups 33 We prove this theorem by induction on | Δ \. For | Δ \ = 1, the assertion is trivial. Let | Δ | > 1. (a) First, we assume that 2| Δ \ < \ Ω |. This implies 2\Γ\ > | Ω |, and therefore for every geG we have ΓηΓ9 Φ 0 . Let α, βΕΔ. 1. Because Γ r\ Γ9 Φ 0 , H = (GA >g~lGAg) is transitive on Γν J \ since GA is transitive on Γ and g~*GAg is transitive on Γ9. Further, Δ = Δ η Δ9 is the set of all points which are left fixed by every element of H. Because a e i we have 1 < | Δ |, and because β \$ Δ we have | Δ \ < \ Δ |.

Every \-fold transitive group is primitive br a Frobenius group. Proof. , let Gx = H have only orbits of length m > 1. Let us assume that G has a block of length k with 1 < k < n = \ Ω |. , n may be arranged in a matrix ai3 in such a way that every row φί is a block of G with length k and, say, « u = 1. φχ is fixed by Hf therefore k = l(w), hence (ky m) = 1. For i > 1, φ? is fixed by / / and does not contain 1. Therefore | φ? | is divisible by my but also by ky therefore by km. On the other hand, | α^ | = m, and therefore | φ?

1. In particular (n — l)-fold transitivity of G follows for m = 2 and (n — 2)-fold transitivity for m = 3 . Hence we have the following theorem. 3. A primitive group which contains a transposition is a symmetric group. A primitive group which contains a 3-cycle is either alternating or symmetric. For certain values of w, there exist groups G of degree n which contain transitive subgroups of degrees < n — 2 and are doubly but not triply transitive. These Jordan groups can be interpreted as automorphism groups of block designs (Hall, 1962).