On perfect isometries and isotypies inalternating groups

Perfect isometries and isotypies are constructed for alternating groups between blocks with abelian defect groups and the Brauer correspondents of these blocks. These perfect isometries and isotypies satisfy additional compatibility conditions which imply that an extended Broué conjecture holds for the principal block of an almost simple group with an abelian Sylow -subgroup and a generalized Fitting subgroup isomorphic to an alternating group.

     

On abelian groups with right-invariant isometries

We describe the reduced periodically complete groups, algebraically compact groups, and torsion-free completely decomposable groups with right-invariant monomorphisms preserving the p-heights of the elements of the group under study.     

Smith equivalence of representations for finite perfect groups

Using smooth one-fixed-point actions on spheres and a result due to Bob Oliver on the tangent representations at fixed points for smooth group actions on disks, we obtain a similar result for perfect group actions on spheres. For a finite group , we compute a certain subgroup of the representation ring . This allows us to prove that a finite perfect group has a smooth -proper action on a sphere with isolated fixed points at which the tangent representations of are mutually nonisomorphic if and only if contains two or more real conjugacy classes of elements not of prime power order. Moreover, by reducing group theoretical computations to number theory, for an integer and primes , we prove similar results for the group , , or . In particular, has Smith equivalent representations that are not isomorphic if and only if , , .

     

On groups with almost perfect involution

The theorems of Jordan, Frobenius, M. Hall, Brauer-Suzuki-Wall, Shunkov, Mazurov, and Belyaev are generalized.     

On element-centralizers in finite groups

For any group G, let |Cent(G)| denote the number of centralizers of its elements. A group G is called n-centralizer if |Cent(G)| = n. In this paper, we find |Cent(G)| for all minimal simple groups. Using these results we prove that there exist finite simple groups G and H with the property that |Cent(G)| = |Cent(H)| but ${G\not\cong H}$ . This result gives a negative answer to a question raised by A. Ashrafi and B. Taeri. We also characterize all finite semi-simple groups G with |Cent(G)| ≤  73.     

On nonsystematic perfect codes over finite fields

The nonsystematic perfect q-ary codes over finite field F _q of length n = (q ^m − 1)/(q − 1) are constructed in the case when m ≥ 4 and q ≥ 2 and also when m = 3 and q is not prime. For q ≠ 3, 5, these codes can be constructed by switching seven disjoint components of the Hamming code H _q ⁿ ; and, for q = 3, 5, eight disjoint components.     

On full-rank perfect codes over finite fields

We propose a construction of full-rank q-ary 1-perfect codes. This is a generalization of the construction of full-rank binary 1-perfect codes by Etzion and Vardy (1994). The properties of the i-components of q-ary Hamming codes are investigated, and the construction of full-rank q-ary 1-perfect codes is based on these properties. The switching construction of 1-perfect codes is generalized to the q-ary case. We propose a generalization of the notion of an i-component of a 1-perfect code and introduce the concept of an (i, σ)-component of a q-ary 1-perfect code. We also present a generalization of the Lindström–Schönheim construction of q-ary 1-perfect codes and provide a lower bound for the number of pairwise distinct q-ary 1-perfect codes of length n.     

On finite tetraprimary groups

A three-player game is considered in which the first and second players have dynamic superiority over the third player. Two fixed time points are specified. The game ends if either the first player captures the third player at the first time point, or the second player captures the third player at the second time point. We analyze a situation when the initial positions in the game are such that neither the first nor the second player alone can capture the third player at the specified points of time. We propose sufficient conditions on the parameters of the game under which, for given initial states of the players, the first and second players by applying some controls can guarantee that one of them will meet the third player at the prescribed moment. Simulation results for a model example are also presented.     

Critical exponents for groups of isometries

Let Γ be a convex co-compact group of isometries of a CAT(−1) space X and let Γ₀ be a normal subgroup of Γ. We show that, provided Γ is a free group, a sufficient condition for Γ and Γ₀ to have the same critical exponent is that Γ / Γ₀ is amenable.      

On automorphism groups of some finite groups

We show that if n > 1 is odd and has no divisor p4 for any prime p, then there is no finite group G such that│Aut(G)│ = n.     

On isometries in GMV-algebras

Let A = (A,⊕,⁻,^∼, 0, 1) be a GMV-algebra and ρ: A × A → A the distance function on A defined by ρ(x, y) = (x∨y)−(x∧y) for each x, y ∈ A.     

Expansion in perfect groups

Let ?? be a subgroup of ${{\rm GL}_d(\mathbb{Z}[1/q_0])}$ generated by a finite symmetric set S. For an integer q, denote by ?? _q the projection map ${\mathbb{Z}[1/q_0] \to \mathbb{Z}[1/q_0]/q \mathbb{Z}[1/q_0]}$ . We prove that the Cayley graphs of ?? _q (??) with respect to the generating sets ?? _q (S) form a family of expanders when q ranges over square-free integers with large prime divisors if and only if the connected component of the Zariski-closure of ?? is perfect, i.e. it has no nontrivial Abelian quotients.     

On permuteral subgroups in finite groups

The permutizer of a subgroup H in a group G is defined as the subgroup generated by all cyclic subgroups of G that permute with H. Call H permuteral in G if the permutizer of H in G coincides with G; H is called strongly permuteral in G if the permutizer of H in U coincides with U for every subgroup U of G containing H. We study the finite groups with given systems of permuteral and strongly permuteral subgroups and find some new characterizations of w-supersoluble and supersoluble groups.