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We classify all finite groups Gthat possess an element x∈ Gsuch that every irreducible character of Gtakes a root of unity value at x.

In this paper, we investigate finite solvable tidy groups. We classify the tidy {p,q}-groups. Combining this with a previous result, we are able to characterize the finite tidy solvable groups. Using this characterization, we bound the Fitting height of finite tidy solvable groups and we prove that the quotients of finite tidy solvable groups are tidy.

We prove that if Gis a finite group, Nis a normal subgroup, and there is a prime pso that all the elements in G∖ Nhave p-power order, then either Gis a p-group or G= PNwhere Pis a Sylow p-subgroup and (G,P,P∩ N) is a Frobenius–Wielandt triple. We also prove that if all the elements of G∖ Nhave prime power orders and the orders are divisible by two primes pand q, then Gis a {p,q}-group and G/Nis either a Frobenius group or a 2-Frobenius group. If all the elements of G∖ Nhave prime power orders and the orders are divisible by at least three primes, then all elements of Ghave prime power order and G/Nis nonsolvable.

A finite group whose irreducible complex non-linear characters are rational is called a Q1-group. In this paper, we study Q1-groups via their Sylow 2-subgroups. In particular, we prove that if Gis a solvable Q1-group that is not a rational group, then its Sylow 2-subgroups are Q1-groups. To prove this, we obtain information about Q1-groups where a Sylow 2-subgroup is contained in the vanishing off subgroup where the vanishing-off subgroup is a proper subgroup. We also obtain results about Q1-groups where a Sylow 2-subgroup is not contained in the vanishing off subgroup. In addition, we prove that Q1-groups whose Sylow 2-subgroups are dihedral and generalized quaternion groups have that all of the irreducible characters are rational groups.

A graph is split if there is a partition of its vertex set into a clique and an independent set. In this paper, we determine when the Gruenberg–Kegel graph, the solvable graph, and the compact forms of these graphs associated with finite nonabelian simple groups are split. In particular, it is proved that the compact form of the Gruenberg–Kegel graph of any finite simple group is split.

When $G$is a finite solvable group, we prove that $|G|$can be bounded by a function in the number of irreducible characters with values in fields where $\mathbb{Q}$is extended by prime power roots of unity. This gives a character theory analog for solvable groups of a theorem of Héthelyi and Külshammer that bounds the order of a finite group in terms of the number of conjugacy classes of elements of prime power order. In particular, we obtain for solvable groups a generalization of Landau’s theorem.

Abstract.For every odd prime and every integer , there is a Heisenberg group of order that has pairwise nonisomorphic quotients of order . Yet, these quotients are virtually indistinguishable. They have isomorphic character tables, every conjugacy class of a non-central element has the same size, and every element has order at most . They are also directly and centrally indecomposable and of the same indecomposability type. Nevertheless, there is a polynomial-time algorithm to test for isomorphisms between these groups.

AbstractWe count the number of lifts of an irreducible π-partial character that lies in a block with a cyclic defect group.

A character pair ( H, θ) in a group G is a subgroup H and a character θ ∈ Irr( H). Following Dade, we say that a character pair ( H, θ) is an inductive source in G if induction to G defines an injective map from the irreducible characters of the stabilizer of ( H, θ) that lie over θ into Irr( G). (By the Clifford correspondence, this necessarily happens if H is normal in G.) A character pair is said to be conjugate stable if its conjugates satisfy a certain technical condition. We show that an inductive source must be conjugate stable and we present an example of a character pair that is conjugate stable but is not an inductive source. Finally, if ( H, θ) is conjugate stable and H is a subnormal subgroup of G, we show that ( H, θ) must be an inductive source in G.

Let G be a group and $\pi$ be a set of primes. We consider the set ${\rm cd}^{\pi}(G)$ of character degrees of G that are divisible only by primes in $\pi$. In particular, we define $\Gamma^{\pi}(G)$ to be the graph whose vertex set is the set of primes dividing degrees in ${\rm cd}^{\pi}(G)$. There is an edge between p and q if pq divides a degree $a \in {\rm cd}^{\pi}(G)$. We show that if G is $\pi$-solvable, then $\Gamma^{\pi}(G)$ has at most two connected components.

Abstract. In this paper, we answer the following question posed by G. Navarro: suppose that S acts on G coprimely, and that A acts on SG so that S and G are both A-invariant. If $\chi $ is an S-invariant irreducible character of G, then must $\chi $ extend to AG if and only if $\chi ^*$ extends to AC_G(S), where $\chi ^*$ is the Glauberman-Isaacs Correspondent for $\chi $ under the action of S? We prove that the answer to this question is yes.

The Birch reduction of (2 -trialkysilyl)furan-3-carboxylic acids has been applied to the synthesis of methyl (2-trialkylsilyl)tetrahydrofuran-3-carboxylates. It is believed that the silicon moiety in such substrates controls the sense of asymmetric induction observed in Michael reaction of the derived enolates with methyl cinnamate. Mild oxidative removal of the silicon moiety generates an oxonium cation which undergoes nucleophilic capture to afford a functionalised furanopyran.

Ozonolysis of a variety of (tetraydrofuran-2-yl)tri-n-butylstannanes affords the corresponding γ-butyrolactones in good to excellent yields. This reaction is tolerant to a range of other functional groups and provides access to substituted γ-butyrolactones not available from aldol reactions of the parent lactone.

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ChemInform is a weekly Abstracting Service, delivering concise information at a glance that was extracted from about 200 leading journals. To access a ChemInform Abstract, please click on HTML or PDF.