Examples of Commutative Rings by Harry C. Hutchins

By Harry C. Hutchins

Booklet in good shape.

Show description

Read Online or Download Examples of Commutative Rings PDF

Similar mathematics_1 books

Arithmétique et travaux pratiques cycle d'observation classe de sixième

Manuel de mathématiques, niveau sixième. Cet ouvrage fait partie de los angeles assortment Lebossé-Hémery dont les manuels furent à l’enseignement des mathématiques ce que le Bled et le Bescherelle furent à celui du français.

Additional resources for Examples of Commutative Rings

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).

Download PDF sample

Rated 4.92 of 5 – based on 15 votes