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• W2077997260 abstract "THEOREM. Let G be afinite group and H a solvable subgroup of G. Suppose that the Schreier conjecture holds. Then G is solvable if G has an H-composition series. Let G be a group and H?G. Let {Gi}o be subnormal series with G,=(1) and Go=G. This series is called an H-composition series if H normalizes each Gi and if there exists no subgroup X properly between G+-1 and G, which is normalized by H. If G is a finite solvable group then for all H_G such H-composition series exist. These can be obtained by refinement into irreducible H-factors of any chief series of G. If G is not solvable then, for particular H, such series may not exist. This is easily seen by letting G be simple nonabelian and H any proper subgroup. The object of this note will be to shed some light on restrictions that one must have on finite groups G and H< G if such H-composition series occur. All groups are finite. If {Gi}n is a subnormal series of G we denote by G(i) the factor G1/Gl and call JG(i)Jn the factors of the series. A factor of G is a group RIS where S < R < G. If KIL is a factor of G then we can in a natural way define AutG(K/L) as N(K)nN(L)/C(K/L) and OutG(KIL) as N(K) nN(L)/KC(K/L). These groups correspond to the automorphisms and outer automorphisms that G induces on the factor KIL. If E is a group, then E is said to be involved in G if E is isomorphic to some factor of G. If E is a nonabelian simple group, then KIL is called a s-factor if it is the direct product of isomorphic copies of E. If E is a nonabelian simple group the Schreier conjecture states that Out(Z)=Aut(Z)/In(Z) is a solvable group. In what follows, if KIL is a simple nonabelian factor of G then if Out6(K/L) is solvable we will say that G satisfies the Schreier conjecture with respect to the factor KIL. Our result is THEOREM. Let H<G with H-composition series {Gi}n. Let z be a nonabelian simple group and G(i) be a E-factor. Suppose G satisfies the Schreier Received by the editors May 30, 1972. AMS (MOS) subject classifications (1970). Primary 20D30. ? American Mathematical Society 1973" @default.
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• W2077997260 date "1973-02-01" @default.
• W2077997260 modified "2023-09-26" @default.
• W2077997260 title "On composition series in finite groups" @default.
• W2077997260 doi "https://doi.org/10.1090/s0002-9939-1973-0311763-1" @default.
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