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1.
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.  相似文献   

2.
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 Dy,t3g(x,y) = 0 \Delta_{y,t}^{3}g(x,y) = 0 and Dx,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 .  相似文献   

3.
4.
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,…,Tk in G such that for each vertex xps of G, the k paths from x to s in T1,…,Tk are pairwise openly disjoint.  相似文献   

5.
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 Cm and Cn be cycles. We prove that f(Cm 2Cn)hf(Cm) f(Cn) 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.  相似文献   

6.
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 QS a projective O G{\cal O} G-lattice such that QS / pQSQ_S / \pi Q_S is a projective cover of S. We show that if S is liftable and QS 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 ).  相似文献   

7.
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.  相似文献   

8.
In this article we determine the irreducible ordinary characters cr \chi_r of a finite group G occurring in a transitive permutation representation (1M )G of a given subgroup M of G, and their multiplicities mr = ((1M)G, cr) 1 0 m_r = ((1_{M})^G, \chi_r) \neq 0 by means of a new explicit formula calculating the coefficients ark of the central idempotents er = ?k=1d ark Dk e_r = \sum\limits_{k=1}^{d} a_{rk} D_k in the intersection algebra B \cal B of (1M )G generated by the intersection matrices Dk corresponding to the double coset decomposition G = èk=1d Mxk M G = \bigcup\limits_{k=1}^{d} Mx_{k} M .¶Furthermore, an explicit formula is given for the calculation of the character values cr(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.  相似文献   

9.
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 mp (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 mp(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 mp(G) £ [(p-1)/(p)] ·|W|m_{p}(G)\le {{p-1}\over{p}} \cdot |\mit\Omega | are identified.  相似文献   

10.
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].  相似文献   

11.
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.  相似文献   

12.
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.  相似文献   

13.
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.  相似文献   

14.
Group Connectivity of 3-Edge-Connected Chordal Graphs   总被引:3,自引:0,他引:3  
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 vf(e)m~e entering vf(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|́.  相似文献   

15.
Let G be a permutation group on a finite set W\Omega . If G does not involve An 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.  相似文献   

16.
I. Levi 《Semigroup Forum》1999,59(3):342-353
For a semigroup S of transformations (total or partial) of a finite n-element set Xn, denote by GS the group of all the permutations h of Xn that preserve S under conjugation. It is shown that, unless S contains certain nilpotents and has a very restricted form, the alternating group Altn may not serve as GS, so that AltnGS implies that GS=Sn, and S is an Sn-normal semigroup.  相似文献   

17.
¶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.  相似文献   

18.
Let n be an integer greater than 1, and let G be a group. A subset {x1, x2, ..., xn} 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¶¶xp(1)xp(2) ?xp(n) = xs(1)xs(2) ?xs(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 Qn 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.  相似文献   

19.
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 {an} be a sequence of real numbers with 0 £ an £ 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 {xn} defined by¶¶xn+1=an x + (1-an) [1/(n+1)] ?j=0n Tj xn   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 \,.  相似文献   

20.
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).  相似文献   

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