Wilson's functional equation on P3-groups

Summary. Consider Wilson's functional equation¶¶f(xy) + f(xy^-1) = 2f(f)g(y) f(xy) + f(xy^{-1}) = 2f(f)g(y) , for f,g : G ? K f,g : G \to K ¶where G is a group and K a field with char K 1 2 {\rm char}\, K\ne 2 .¶Aczél, Chung and Ng in 1989 have solved Wilson's equation, assuming that the function g satisfies Kannappan's condition g(xyz) = g(xzy) and f(xy) = f(yx) for all x,y,z ? G x,y,z\in G .¶In the present paper we obtain the general solution of Wilson's equation when G is a P₃-group and we show that there exist solutions different of those obtained by Aczél, Chung and Ng.¶A group G is said to be a P₃-group if the commutator subgroup G' of G, generated by all commutators [x,y] := x^-1y^-1xy, has the order one or two.     

A wavelike functional equation of Pexider type

Summary. Let (G, +) and (H, +) be abelian groups such that the equation 2u = v 2u = v is solvable in both G and H. It is shown that if f₁, f₂, f₃, f₄, : G ×G ? H f_1, f_2, f_3, f_4, : G \times G \longrightarrow H satisfy the functional equation f₁(x + t, y + s) + f₂(x - t, y - s) = f₃(x + s, y - t) + f₄(x - s, y + t) for all x, y, s, t ? G x, y, s, t \in G , then f₁, f₂, f₃, and f₄ are given by f₁ = w + h, f₂ = w - h, f₃ = w + k, f₄ = w - k where w : G ×G ? H w : G \times G \longrightarrow H is an arbitrary solution of f (x + t, y + s) + f (x - t, y - s) = f (x + s, y - t) + f (x - s, y + t) for all x, y, s, t ? G x, y, s, t \in G , and h, k : G ×G ? H h, k : G \times G \longrightarrow H are arbitrary solutions of D_y,t³g(x,y) = 0 \Delta_{y,t}^{3}g(x,y) = 0 and D_x,t³g(x,y) = 0 \Delta_{x,t}^{3}g(x,y) = 0 for all x, y, s, t ? G x, y, s, t \in G .     

Independent Trees in Planar Graphs Independent trees

We prove the following statement: Let G be a finite k-connected undirected planar graph and s be a vertex of G. Then there exist k spanning trees T₁,…,T_k in G such that for each vertex xps of G, the k paths from x to s in T₁,…,T_k are pairwise openly disjoint.     

The 2-Pebbling Property and a Conjecture of Graham's

The pebbling number of a graph G, f(G), is the least m such that, however m pebbles are placed on the vertices of G, we can move a pebble to any vertex by a sequence of moves, each move taking two pebbles off one vertex and placing one on an adjacent vertex. It is conjectured that for all graphs G and H, f(G 2H)hf(G)f(H).¶Let C_m and C_n be cycles. We prove that f(C_m 2C_n)hf(C_m) f(C_n) for all but a finite number of possible cases. We also prove that f(G2T)hf(G) f(T) when G has the 2-pebbling property and T is any tree.     

On standard Auslander-Reiten sequences for finite groups

Let G be a finite group and O{\cal O} a complete discrete valuation ring of characteristic zero with maximal ideal (p)(\pi ) and residue field k = O/(p)k = {\cal O}/(\pi ) of characteristic p > 0. Let S be a simple kG-module and Q_S a projective O G{\cal O} G-lattice such that Q_S / pQ_SQ_S / \pi Q_S is a projective cover of S. We show that if S is liftable and Q_S belongs to a block of O G{\cal O} G of infinite representation type, then the standard Auslander-Reiten sequence terminating in W^-1S\Omega ^{-1}S is a direct summand of the short exact sequence obtained from some Auslander-Reiten sequence of OG{\cal O}G-lattices by reducing each term mod (p)(\pi ).     

Every Invariant Measure Semigroup Contains an Ideal which is Embeddable in a Group

We show that every invariant measure semigroup S with associated invariant measure mu contains an ideal S₀ which is embeddable as an open subsemigroup in a locally compact abelian group G in such a way that the restriction to S₀ of mu coincides with the restriction to S₀ of a Haar measure on G. This is a positive answer to a question posed by J.H Williamson. As a consequence the generalization of Pontryagin's duality theorem for S is obtained.     

The character values of the irreducible constituents of a transitive permutation representation

In this article we determine the irreducible ordinary characters c_r \chi_r of a finite group G occurring in a transitive permutation representation (1_M )^G of a given subgroup M of G, and their multiplicities m_r = ((1_M)^G, c_r) 1 0 m_r = ((1_{M})^G, \chi_r) \neq 0 by means of a new explicit formula calculating the coefficients a_rk of the central idempotents e_r = ?_k=1^d a_rk D_k e_r = \sum\limits_{k=1}^{d} a_{rk} D_k in the intersection algebra B \cal B of (1_M )^G generated by the intersection matrices D_k corresponding to the double coset decomposition G = è_k=1^d Mx_k M G = \bigcup\limits_{k=1}^{d} Mx_{k} M .¶Furthermore, an explicit formula is given for the calculation of the character values c_r(x) \chi_{r}(x) of each element x ? G x \in G . Using this character formula we obtain a new practical algorithm for the calculation of a substantial part of the character table of G.     

On minimal p-degrees in 2-transitive permutation groups

Suppose G is a transitive permutation group on a finite set W\mit\Omega of n points and let p be a prime divisor of |G||G|. The smallest number of points moved by a non-identity p-element is called the minimal p-degree of G and is denoted m_p (G). ¶ In the article the minimal p-degrees of various 2-transitive permutation groups are calculated. Using the classification of finite 2-transitive permutation groups these results yield the main theorem, that m_p(G) 3 [(p-1)/(p+1)] ·|W|m_{p}(G) \geq {{p-1} \over {p+1}} \cdot |\mit\Omega | holds, if Alt(W) \nleqq G {\rm Alt}(\mit\Omega ) \nleqq G .¶Also all groups G (and prime divisors p of |G||G|) for which m_p(G) ￡ [(p-1)/(p)] ·|W|m_{p}(G)\le {{p-1}\over{p}} \cdot |\mit\Omega | are identified.     

The fixed points of coprime action

Let P be an odd p\pi -group that acts as a group of automorphisms on the soluble p￠\pi '-group G. We obtain generators for the fixed points of P on [G, P].     

On a filtered multiplicative basis of group algebras

Let K be a field of characteristic p and G a nonabelian metacyclic finite p-group. We give an explicit list of all metacyclic p-groups G, such that the group algebra KG over a field of characteristic p has a filtered multiplicative K-basis. We also present an example of a non-metacyclic 2-group G, such that the group algebra KG over any field of characteristic 2 has a filtered multiplicative K-basis.     

On a certain ideal of Külshammer in the centre of a group algebra

Let G be a finite group and let F be a splitting field of characteristic $ p > 0 $ p > 0 . We show that I² = E₀, where I is a certain ideal of the centre Z of FG, and E₀ is the span of the block idempotents of defect zero.     

On finitary linear groups with a locally nilpotent maximal subgroup

We prove that a group G of finitary permutations, containing a locally nilpotent maximal subgroup M is locally solvable if M is not a 2-group. We also prove that the same is true if G is a periodic, non-modular, finitary linear group.     

Group Connectivity of 3-Edge-Connected Chordal Graphs

Let A be a finite abelian group and G be a digraph. The boundary of a function f: E(G)ZA is a function ‘f: V(G)ZA given by ‘f(v)=~_{e leaving v}f(e)m~_{e entering v}f(e). The graph G is A-connected if for every b: V(G)ZA with ~_{v] V(G)} b(v)=0, there is a function f: E(G)ZA{0} such that ‘f=b. In [J. Combinatorial Theory, Ser. B 56 (1992) 165-182], Jaeger et al showed that every 3-edge-connected graph is A-connected, for every abelian group A with |A|̈́. It is conjectured that every 3-edge-connected graph is A-connected, for every abelian group A with |A|̓ and that every 5-edge-connected graph is A-connected, for every abelian group A with |A|́.¶ In this note, we investigate the group connectivity of 3-edge-connected chordal graphs and characterize 3-edge-connected chordal graphs that are A-connected for every finite abelian group A with |A|́.     

Orbits of permutation groups on the power set

Let G be a permutation group on a finite set W\Omega . If G does not involve A_n for n \geqq 5 n \geqq 5 , then there exist two disjoint subsets of W\Omega such that no Sylow subgroup of G stabilizes both and four disjoint subsets of W\Omega whose stabilizers in G intersect trivially.     

On Groups Associated with Transformation Semigroups

For a semigroup S of transformations (total or partial) of a finite n-element set X_n, denote by G_S the group of all the permutations h of X_n that preserve S under conjugation. It is shown that, unless S contains certain nilpotents and has a very restricted form, the alternating group Alt_n may not serve as G_S, so that Alt_n ⊆ G_S implies that G_S=S_n, and S is an S_n-normal semigroup.     

Generalized polygons with split Levi factors

¶Theorem. Suppose that the group G has an irreducible spherical BN-pair of rank 2. If for every 2-transitive subgroup L of the Levi factors of G the one-point stabilizer B splits as B = U 2 T with [B, B] = U, then the Tits-building associated to the BN-pair of G is Moufang.¶¶This implies in particular that a simple group G with a spherical BN-pair of rank 2 whose Levi factors are permutation equivalent to subgroups of PGL₂(K) or the Suzuki group GSz(K) is (essentially) classical.     

On <Emphasis Type="Italic">T</Emphasis>-groups,supersolvable groups,and maximal subgroups

A group is called a T-group if all its subnormal subgroups are normal. Finite T-groups have been widely studied since the seminal paper of Gaschütz (J. Reine Angew. Math. 198 (1957), 87–92), in which he described the structure of finite solvable T-groups. We call a finite group G an NNM-group if each non-normal subgroup of G is contained in a non-normal maximal subgroup of G. Let G be a finite group. Using the concept of NNM-groups, we give a necessary and sufficient condition for G to be a solvable T-group (Theorem 1), and sufficient conditions for G to be supersolvable (Theorems 5, 7 and Corollary 6).     

A strong convergence theorem for asymptotically nonexpansive mappings in Banach spaces

Let C be a closed, convex subset of a uniformly convex Banach space whose norm is uniformly Gâteaux differentiable and let T be an asymptotically nonexpansive mapping from C into itself such that the set F (T) of fixed points of T is nonempty. Let {a_n} be a sequence of real numbers with 0 ￡ a_n ￡ 10 \leq a_n \leq 1, and let x and x₀ be elements of C. In this paper, we study the convergence of the sequence {x_n} defined by¶¶x_n+1=a_n x + (1-a_n) [1/(n+1)] ?_j=0ⁿ T^j x_n   x_{n+1}=a_n x + (1-a_n) {1\over n+1} \sum\limits_{j=0}^n T^j x_n\quad for n=0,1,2,...  . n=0,1,2,\dots \,.     

Odd order groups with the rewriting property Q3

Let n be an integer greater than 1, and let G be a group. A subset {x₁, x₂, ..., x_n} of n elements of G is said to be rewritable if there are distinct permutations p \pi and s \sigma of {1, 2, ..., n} such that¶¶x_p(1)x_p(2) ?x_p(n) = x_s(1)x_s(2) ?x_s(n). x_{\pi(1)}x_{\pi(2)} \ldots x_{\pi(n)} = x_{\sigma(1)}x_{\sigma(2)} \ldots x_{\sigma(n)}. ¶¶A group is said to have the rewriting property Q_n if every subset of n elements of the group is rewritable. In this paper we prove that a finite group of odd order has the property Q₃ if and only if its derived subgroup has order not exceeding 5.