Suppose that AG is a solvable group with normal subgroup G where (|A|, |G|) = 1. Assume that A is a class two odd p group all of whose irreducible representations are isomorphic to subgroups of extra special p groups. If pc \u2260 rd + 1 for any c = 1, 2 and any prime r where r2d+1 divides |G| and if CG(A) = 1 then the Fitting length of G is bounded by the power of p dividing |A|.
\r\n\r\nThe theorem is proved by applying a fixed point theorem to a reduction of the Fitting series of G. The fixed point theorem is proved by reducing a minimal counter example. IF R is an extra spec r subgroup of G fixed by A1, a subgroup of A, where A1 centralizes D(R), then all irreducible characters of A1R which are nontrivial on Z(R) are computed. All nonlinear characters of a class two p group are computed.
\r\n", "doi": "10.7907/NYAT-FG53", "publication_date": "1967", "thesis_type": "phd", "thesis_year": "1967" }, { "id": "thesis:9176", "collection": "thesis", "collection_id": "9176", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:09252015-104533327", "type": "thesis", "title": "Rings Faithfully Represented on their Left Socle", "author": [ { "family_name": "Gordon", "given_name": "Robert", "clpid": "Gordon-Robert" } ], "thesis_advisor": [ { "family_name": "Dade", "given_name": "Everett C.", "clpid": "Dade-E-C" } ], "thesis_committee": [ { "family_name": "Unknown", "given_name": "Unknown" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "In 1964 A. W. Goldie [1] posed the problem of determining all\r\nrings with identity and minimal condition on left ideals which are\r\nfaithfully represented on the right side of their left socle. Goldie\r\nshowed that such a ring which is indecomposable and in which the left\r\nand right principal indecomposable ideals have, respectively, unique\r\nleft and unique right composition series is a complete blocked\r\ntriangular matrix ring over a skewfield. The general problem\r\nsuggested above is very difficult. We obtain results under certain\r\nnatural restrictions which are much weaker than the restrictive\r\nassumptions made by Goldie.
\r\n\r\nWe characterize those rings in which the principal indecomposable\r\nleft ideals each contain a unique minimal left ideal (Theorem (4.2)). It\r\nis sufficient to handle indecomposable rings (Lemma (1.4)). Such a\r\nring is also a blocked triangular matrix ring. There exist r positive\r\nintegers K1,..., Kr such that the i,jth block of a typical matrix is a\r\nKi x Kj matrix with arbitrary entries in a subgroup Dij of the additive group of a fixed skewfield D. Each Dii is a sub-skewfield of D and Dri = D for all i. Conversely, every matrix ring which has this form is\r\nindecomposable, faithfully represented on the right side of its left socle,\r\nand possesses the property that every principal indecomposable left ideal\r\ncontains a unique minimal left ideal.
\r\n\r\nThe principal indecomposable left ideals may have unique composition\r\nseries even though the ring does not have minimal condition on\r\nright ideals. We characterize this situation by defining a partial ordering\r\n\u03c1 on {i, 2,...,r} where we set i\u03c1j if Dij \u2260 0. Every principal indecomposable\r\nleft ideal has a unique composition series if and only if the\r\ndiagram of \u03c1 is an inverted tree and every Dij is a one-dimensional left\r\nvector space over Dii (Theorem (5.4)).
\r\n\r\nWe show (Theorem (2.2)) that every ring A of the type we are\r\nstudying is a unique subdirect sum of less complex rings A1,...,As\r\nof the same type. Namely, each Ai has only one isomorphism class\r\nof minimal left ideals and the minimal left ideals of different Ai are\r\nnon-isomorphic as left A-modules. We give (Theorem (2.1))\r\nnecessary and sufficient conditions for a ring which is a subdirect sum\r\nof rings Ai having these properties to be faithfully represented on the\r\nright side of its left socle. We show ((4.F), p. 42) that up to technical\r\ntrivia the rings Ai are matrix rings of the form
\r\n\r\n\r\n\r\n[...]. Each Qj comes from the faithful irreducible \r\nmatrix representation of a certain skewfield over a fixed skewfield D.\r\nThe bottom row is filled in by arbitrary elements of D.
\r\n\r\nIn Part V we construct an interesting class of rings faithfully\r\nrepresented on their left socle from a given partial ordering on a\r\nfinite set, given skewfields, and given additive groups. This class of\r\nrings contains the ones in which every principal indecomposable left\r\nideal has a unique minimal left ideal. We identify the uniquely\r\ndetermined subdirect summands mentioned above in terms of the given\r\npartial ordering (Proposition (5.2)). We conjecture that this technique\r\nserves to construct all the rings which are a unique subdirect sum of\r\nrings each having the property that every principal-indecomposable left \r\nideal contains a unique minimal left ideal.
", "doi": "10.7907/1S2B-8413", "publication_date": "1966", "thesis_type": "phd", "thesis_year": "1966" } ]