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In this paper, we study the regular solvability (in Sobolev spaces) of transmission problems for parabolic second-order systems with conjugation conditions of non-ideal contact type. The solution of such a problem has all generalized derivatives entering in the equation that are summable with some power $p\in (1,\infty)$ . One can express limit values of conormal derivatives at the interface in terms of combinations of limit values of the solution. This problem, arising when describing heat and mass transfer processes, differs from the classical statement of diffraction problems. The proof relies on derived a priori estimates and on the method of continuation in a parameter.

Let π be an arbitrary set of primes. A very broad generalization of the notion of nilpotent group is the notion of π-decomposable group, which is the direct product of a π-group and a π′-group. In this paper, we obtain a description of finite π-indecomposable groups all of whose 2-maximal subgroups are π-decomposable. The proof involves the author’s recent results connected with the notion of control of the prime spectrum of a finite simple group. Finite nonnilpotent groups all of whose 2-maximal subgroups are nilpotent were studied by Z. Janko in 1962 in the case of nonsolvable groups and by the author in 1968 in the case of solvable groups.

The set π(G) of all prime divisors of the order of a finite group G is often called its prime spectrum. It is proved that every finite simple nonabelian group G has sections H 1, …, H m of some special form such that π(H 1)∪…∪π(H m ) = π(G) and m ≤ 5. Moreover, m ≤ 2 if G is an alternating or classical simple group. In all cases, it is possible to choose the sections H i so that each of them is a simple nonabelian group, a Frobenius group, or (in one case) a dihedral group. If the above equality holds for a finite group G, then we say that the set {H 1,…,H m } controls the prime spectrum of G. We also study some parameter c(G) of finite groups G related to the notion of control.

The investigation of nonclassical interactions (orthogonality relations) in finite groups is continued. A description of all small interactions (small D-blocks and small Φ-blocks) in symplectic groups Sp4(q) for even q is obtained. In particular, two known conjectures on these interactions are confirmed for these groups. Note that the existence of a small D-block in a group is equivalent to the existence of a pair of semiproportional rows in its character table and the existence of a small Φ-block in a group is equivalent to the existence of a pair of semiproportional columns in its character table. © 2012 Pleiades Publishing, Ltd.

In the author’s previous paper, the hypothesis that the alternating groups An have no pairs of semiproportional irreducible characters is reduced to a hypothesis concerning the problem of describing the pairs of irreducible characters of the symmetric group Sn that are semiproportional on one of the sets An or Sn∖An. In this hypothesis, properties of such a pair of characters are expressed in terms of Young’s diagrams corresponding to these characters. The theorem proved in this paper allows one to exclude from consideration some stages of the verification of this hypothesis.

In studying the pairs of irreducible characters of the symmetric group Sn with the same zero set on An or Sn ∖ An (as well as the pairs of irreducible characters with the same zero set on the alternating group An), the results are important on the connection between the Young diagrams of the characters of these pairs. We prove a theorem that considerably generalizes two previous results of frequent use in this direction.

All finite Young diagrams not containing hooks of length 4 are found. Self-associated diagrams possessing this property are subdivided into three series. Sets of all hook lengths are determined for diagrams contained in each series. The research conducted has proven necessary for the study of certain pairs of irreducible characters of symmetric and alternating groups.

Previously, we dubbed the conjecture that the alternating group An has no semiproportional irreducible characters for any natural n [1]. This conjecture was then shown to be equivalent to the following [3]. Let α and β be partitions of a number n such that their corresponding characters χα and χβ in the group Sn are semiproportional on An. Then one of the partitions α or β is self-associated. Here, we describe all pairs (α, β) of partitions satisfying the hypothesis and the conclusion of the latter conjecture.

The investigation of the pairs of irreducible characters of the symmetric group Sn that have the same set of roots in one of the sets An and Sn ∖ An is continued. All such pairs of irreducible characters of the group Sn are found in the case when the least of the main diagonal’s lengths of the Young diagrams corresponding to these characters does not exceed 2. Some arguments are obtained for the conjecture that alternating groups An have no pairs of semiproportional irreducible characters.

We show that treating of (non-trivial) pairs of irreducible characters of the group Sn sharing the same set of roots on one of the sets An and Sn \ An is divided into three parts. This, in particular, implies that any pair of such characters ?a and ?s (a and s are respective partitions of a number n) possesses the following property: lengths d(a) and d(s) of principal diagonals of Young diagrams for a and s differ by at most 1.

Two characters of a finite group G are semiproportional if they are not proportional and G is a union of two disjoint normal subsets such that the restrictions of these characters to each of the subsets are proportional. We obtain some results on the structure of an arbitrary finite group having a pair of semiproportional irreducible characters; in particular, assertions on the order of the group and on the kernels of semiproportional characters. We also consider the following conjecture: Semiproportional irreducible characters of a finite group have equal degrees. We validate this conjecture for 2-decomposable groups and prove that if the conjecture holds for two groups then it holds for their direct product.

In the representation theory of symmetric groups, for each partition α of a natural number n, the partition h(α) of n is defined so as to obtain a certain set of zeros in the table of characters for Sn. Namely, h(α) is the greatest (under the lexicographic ordering ≤) partition among β ∈ P(n) such that χα(gβ) ≠ 0. Here, χα is an irreducible character of Sn, indexed by a partition α, and gβ is a conjugacy class of elements in Sn, indexed by a partition β. We point out an extra set of zeros in the table that we are dealing with. For every non self-associated partition α ∈ P(n), the partition f(α) of n is defined so that f(α) is greatest among the partitions β of n which are opposite in sign to h(α) and are such that χα(gβ) ≠ 0 (Thm. 1). Also, for any self-associated partition α of n > 1, we construct a partition $$\tilde f$$ (α) ∈ P(n) such that $$\tilde f$$ (α) is greatest among the partitions β of n which are distinct from h(α) and are such that χα(gβ) ≠ 0 (Thm. 2).

Two characters ϕ and ψ of a finite group G are called semiproportional if they are not proportional and there exists a set M in G such that the restrictions of ϕ and ψ to M and G∖M are proportional. We obtain a description for all pairs of proportional irreducible characters of symmetric groups. Namely, in Theorem 1 we prove equivalence of the following conditions for a pair (ϕ,ψ) of different irreducible characters of Sn (n∈ N): (1) ϕ and ψ are semiproportional; (2) ϕ and ψ have the same roots; and (3) ϕ are ψ are associated (i.e., ψ=ϕξ where ξ is a linear character of Sn with kernel An). Note that (1) and (2) are in general not equivalent for arbitrary finite groups. For the symmetric groups, the equivalence of (1) and (3) validates the following conjecture proven earlier by the author for a number of group classes: semiproportional irreducible characters of a finite group have the same degree. The alternating groups seem to have no semiproportional irreducible characters. Theorem 2 of this article is a step towards proving this conjecture.

We argue that an erased row or column in a table of characters of a finite group can be constructively recovered from retained data.

A character table X of a finite group is broken up into four squares: A, B, C, and D. We establish relations via which ranks of the matrices inX are connected. In particular, ifX is an l × l-matrix, A is an s × t-matrix, and, moreover, the squares A and C are opposite, thenr(A)=r(C) + s + t − l; here.r(M) is the rank of a matrix M. Associated with such each block ofX is some integral nonnegative parameter m, and we have m=0 iff A, B, C, and D are active fragments ofX.

In this article, properties of the biprimary subgroups of even order are used to characterize the following series of finite simple groups: the projective special linear groups PSL(2, q), q > 3, the projective special unitary groups PSU(3, 22n), n > 1, the Suzuki groups Sz(q), the Janko group J1 of order 175560, and the Ree-type groups.

The properties of even-order doubly primary subgroups are used to characterize the Ree-type groups as well as certain other finite simple groups.

A property of the groups PSL (2, 2n) and Sz(q) is derived which no other finite simple group possesses. This property restricts the structure of biprimary subgroups of the group. A description is given of all finite nonsolvable groups with this property.

The paper studies the structure of finite groups in which, for any biprimary subgroup B, eitherl2(B) ≤ 1 or O2(B) is a metacyclic group. As a corollary of the result obtained here and of known results of other authors, a description is adduced of finite simple groups in which the intersection of any two distinct Sylow 2-subgroups is metacyclic.