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11.
Brent Kerby 《代数通讯》2013,41(12):5087-5103
In 1993, Muzychuk [23 Muzychuk , Mikhail E. ( 1993 ). The structure of rational Schur rings over cyclic groups . European Journal of Combinatorics 14 : 479490 .[Crossref], [Web of Science ®] [Google Scholar]] showed that the rational Schur rings over a cyclic group Z n are in one-to-one correspondence with sublattices of the divisor lattice of n, or equivalently, with sublattices of the lattice of subgroups of Z n . This can easily be extended to show that for any finite group G, sublattices of the lattice of characteristic subgroups of G give rise to rational Schur rings over G in a natural way. Our main result is that any finite group may be represented as the (algebraic) automorphism group of such a rational Schur ring over an abelian p-group, for any odd prime p. In contrast, over a cyclic group the automorphism group of any Schur ring is abelian. We also prove a converse to the well-known result of Muzychuk [24 Muzychuk , Mikhail E. ( 1994 ). On the structure of basic sets of Schur rings over cyclic groups . Journal of Algebra 169 : 655678 .[Crossref], [Web of Science ®] [Google Scholar]] that two Schur rings over a cyclic group are isomorphic if and only if they coincide; namely, we show that over a group which is not cyclic, there always exist distinct isomorphic Schur rings.  相似文献   
12.
We classify two types of finite groups with certain normality conditions, namely SSN groups and groups with all noncyclic subgroups normal. These conditions are key ingredients in the study of the multiplicative Jordan decomposition problem for group rings.  相似文献   
13.
14.
Let K be a field of characteristic p > 0, let L be a restricted Lie algebra and let R be an associative K-algebra. It is shown that the various constructions in the literature of crossed product of R with u(L) are equivalent. We calculate explicit formulae relating the parameters involved and obtain a formula which hints at a noncommutative version of the Bell polynomials.  相似文献   
15.
In this article a class of subgroups of a finite group G, called Q-injectors, is introduced. If G is soluble, the Q-injectors are precisely the injectors of the Fitting sets. A characterization of nilpotent Q-injectors is given as well as a sufficient condition for the solubility of a finite group G, in terms of Q-injectors, which generalizes a well known result.  相似文献   
16.
S. Ya. Grinshpon 《代数通讯》2013,41(11):4273-4282
We study primary groups containing proper fully invariant subgroups isomorphie to the group. The admissable sequence of Ulm–Kaplansky invariants for primary groups is introduced. Using these sequences, If-groups are described in the classes of seperable and periodically complete groups.  相似文献   
17.
Let W be a finite Coxeter group, P a parabolic subgroup of W, and N W (P) the normalizer of P in W. We prove that every element in N W (P) is strongly real in N W (P), and that every irreducible complex character of N W (P) has Frobenius-Schur indicator 1.  相似文献   
18.
Following Rose, a subgroup H of a group G is called contranormal, if G = H G . In certain sense, contranormal subgroups are antipodes to subnormal subgroups. It is well known that a finite group is nilpotent if and only if it has no proper contranormal subgroups. However, for the infinite groups this criterion is not valid. There are examples of non-nilpotent infinite groups whose subgroups are subnormal; in paricular, these groups have no contranormal subgroups. Nevertheless, for some classes of infinite groups, the absence of contranormal subgroups implies the nilpotency of the group. The current article is devoted to the search of such classes. Some new criteria of nilpotency in certain classes of infinite groups have been established.  相似文献   
19.
《代数通讯》2013,41(5):2357-2379
Abstract

Restrictions of irreducible representations of classical algebraic groups to root A 1-subgroups, i.e., subgroups of type A 1 generated by root subgroups associated with two opposite roots, are studied. Composition factors of such restrictions are found in the following cases: for groups of types A n with n > 2 and D n , for groups of type B n , n > 2, and long root subgroups, for groups of type C n , n > 2, and short root subgroups, and for p-restricted representations of A 2(K), C 2(K) (recall that B 2(K) ? C 2(K)), and of B n (K), n > 2, and short root subgroups. Here we assume that p > 2 for G = B n (K) or C n (K).  相似文献   
20.
《代数通讯》2013,41(7):3471-3486
Abstract

Taking G to be a Chevalley group of rank at least 3 and U to be the unipotent radical of a Borel subgroup B,an extremal subgroup A is an abelian normal subgroup of U which is not contained in the intersection of all the unipotent radicals of the rank 1 parabolic subgroups of G containing B. If there is an unique rank 1 parabolic subgroup P of G containing B with the property that A is not contained in the unipotent radical of P,then A is called a unique node extremal subgroup. In this paper we investigate the embedding of unique node extremal subgroups in U and prove that,apart from some specified cases,such a subgroup is contained in the unipotent radical of a certain maximal parabolic subgroup.  相似文献   
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