On Finite Number of Conjugacy Classes in Groups

The work is inspired by an article of Herzog, Longobardi, and Maj, who considered groups with a finite number of infinite conjugacy classes. Their main results were obtained under assumption that the FC-center is of finite index in the group. We consider here infinite groups with a finite number of conjugacy classes of any size (FNCC-groups). Hence the FC-center in our case will be finite, but of infinite index in the group. Among results on these groups we give a criterion for a wreath product of FNCC-groups to be an FNCC-group.     

On the Average Number of Sylow Subgroups in Finite Groups

Czechoslovak Mathematical Journal - We prove that if the average number of Sylow subgroups of a finite group is less than $${{41} over 5}$$ and not equal to $${{29} over 4}$$ , then G is solvable...     

On Finite Groups with a Given Number of Centralizers

For a finite group G, let Cent(G) denote the set of centralizers of single elements of G and #Cent(G) = |Cent(G)|. G is called an n-centralizer group if #Cent(G) = n, and a primitive n-centralizer group if #Cent(G) = #Cent(G/Z(G)) = n. In this paper, we compute #Cent(G) for some finite groups G and prove that, for any positive integer n 2, 3, there exists a finite group G with #Cent(G) = n, which is a question raised by Belcastro and Sherman [2]. We investigate the structure of finite groups G with #Cent(G) = 6 and prove that, if G is a primitive 6-centralizer group, then G/Z(G) A₄, the alternating group on four letters. Also, we prove that, if G/Z(G) A₄, then #Cent(G) = 6 or 8, and construct a group G with G/Z(G) A₄ and #Cent(G) = 8.This research was in part supported by a grant from IPM.2000 Mathematics Subject Classification: 20D99, 20E07     

Composition Factors of the Finite Groups Isospectral to Simple Classical Groups

Siberian Mathematical Journal - Isospectral are the groups with coinciding sets of element orders. We prove that no finite group isospectral to a finite simple classical group has the...     

Circularity of Finite Groups without Fixed Points

Let áA, B | A^m=1, Bⁿ=A^t, BAB^-1=A^r?\langle A, B\,\vert\, A^\mu=1,\, B^\nu=A^t,\, BAB^{-1}=A^\rho\rangle where and are relative prime numbers, t = /s and s = gcd( – 1,), and is the order of modulo . We prove that if (1) = 2, and (2) is embeddable into the multiplicative group of some skew field, then is circular. This means that there is some additive group N on which acts fixed point freely, and |((a)+b)((c)+d)| 2 whenever a,b,c,d N, a0c, are such that (a)+b(c)+d.     

Circularity of Finite Groups without Fixed Points

Let be a fixed point free group given by the presentation where and are relative prime numbers, t = /s and s = gcd( – 1,), and is the order of modulo . We prove that if (1) = 2, and (2) is embeddable into the multiplicative group of some skew field, then is circular. This means that there is some additive group N on which acts fixed point freely, and |((a)+b)((c)+d)| 2 whenever a,b,c,d N, a0c, are such that (a)+b(c)+d.     

有限环上典型群的阶

本文通过“取模”，“取值和”，“取积”等方法，将有单位元的有限环Ｒ上典型群阶的计算转化为有限域上典型群阶的计算，并计算了Ｒ上ｎ－维自由模Ｖ_ｎ（Ｒ）中ｋ－维自由子模的个数．     

Pro-p Groups with Finite Number of Ends

The number of ends for a pro-p group is defined. The adequacy of the definition is confirmed by the obtained pro-p analogs of results on the number of ends of abstract groups. In particular, it is shown that, as in the abstract case, a pro-p group can have only 0, 1, 2, or infinitely many ends; pro-p groups with two ends are classified and a sufficient condition for a pro-p group to have precisely one end is obtained.     

On the Average Number of Unitary Factors of Finite Abelian Groups (II)

Let t(G) be the number of unitary factors of finite abelian group G. In this paper we prove T(x)=∑ _|G|≤x t(G) = main terms for any exponent pair (κ1/2 + 2κ), which improves on the exponent 9/25 obtained by Xiaodong Cao and the author. Received December 8, 1998, Revised April 27, 1998, Accepted June 12, 1998     

关于有限阿贝尔群单因子个数的均值(Ⅱ)

令ｔ（Ｇ）为有限阿贝尔群Ｇ的单因子个数．本文得到了∑｜Ｇ｜≤ｘｔ（Ｇ）带有更好余项的渐近公式．     

Derangements and Eigenvalue-Free Elements in Finite Classical Groups

Permutations that have no fixed points have been known for avery long time as ‘derangements’. Under that headingRouse-Ball [10, p. 46] puts the matter in the following charmingway: ‘Suppose you have written a letter to each of n differentfriends, and addressed the n corresponding envelopes. In howmany ways can you make the regrettable mistake of putting everyletter into a wrong envelope?’ He traces this problemback to 1713 and since then it has occurred in one form or anotherin many elementary texts on probability theory (see for example[12, p. 57] and references cited by Rouse-Ball). Let p_n be theprobability that all the letters are put into the wrong envelope.It is well known that p_ne^–1 as n, and that, moreover,convergence is very fast. In fact, , so that . A derangement can be thought of as a fixed-point-free elementof the symmetric group Sym(n). In this paper we turn our attentionto eigenvalue-free elements of finite linear groups. Since aneigenvector of a linear transformation X in a vector space Vcorresponds to a fixed point of X in the projective space whosepoints are the 1-dimensional subspaces of V, eigenvalue-freeinvertible matrices correspond to derangements in projectivegroups.     

有限交换环上典型群的Sylow子群

令Ｒ为有限交换局部环，Ｍ表其唯一的极大理想，ｋ表其剩余类域．本文定出了Ｒ上的一般线性群ＧＬｎＲ，辛群Ｓｐ２ｎＲ及双曲正交群Ｏ２ｎＲ的Ｓｙｌｏｗ子群．一般讲，若ｃｈａｒｘ＝ｐ，上述三类典型群的Ｓｙｌｏｗ ｐ－子群分别同构于由某些特殊形式的矩阵生成的子群；若ｃｈａｒｋ≠ｐ，上述三类典型群的Ｓｙｌｏｗｐ－子群分别同构于一循环群或半二面体群与若干Ｚｐ型循环群的圈积。     

有限交换环上典型群的Carter子群

令R为有限交换局部环,K为其剩余类域,令|K|=q.本文研究了R上辛群Sp2nR和正交群O2nR的Carter子群的存在性及结构,并给出R上正交群O2nR在q≡-1（mod 4）情况下的Sylow 2-子群的正确描述.     

On the Heyde Theorem for Finite Abelian Groups

It is well-known Heyde's characterization theorem for the Gaussian distribution on the real line: if _j are independent random variables, _j , _j are nonzero constants such that _i ± _j ^–1 _j 0 for all i j and the conditional distribution of L ₂=_{1 1} + ··· + _{n n} given L ₁=_{1 1} + ··· + _{n n} is symmetric, then all random variables _j are Gaussian. We prove some analogs of this theorem, assuming that independent random variables take on values in a finite Abelian group X and the coefficients _j , _j are automorphisms of X.     

On the Probabilistic Zeta Function for Finite Groups

LetGbe a finite group, and define the function[formula]where μ is the Möbius function on the subgroup lattice ofG. The functionP(G, s) is the multiplicative inverse of a zeta function forG, as described by Mann and Boston. Boston conjectured thatP′(G, 1) = 0 ifGis a nonabelian simple. We will prove a generalization of this conjecture, showing thatP′(G, 1) = 0 unlessG/O_p(G) is cyclic for some primep.     

On Finite n-Acentralizer Groups

Let G be a group and Aut(G) be the group of automorphisms of G. Then the Acentralizer of an automorphism α ∈Aut(G) in G is defined as C _G (α) = {g ∈ G∣α(g) = g}. For a finite group G, let Acent(G) = {C _G (α)∣α ∈Aut(G)}. Then for any natural number n, we say that G is n-Acentralizer group if |Acent(G)| =n. We show that for any natural number n, there exists a finite n-Acentralizer group and determine the structure of finite n-Acentralizer groups for n ≤ 5.     

Parker Vectors for Infinite Groups

We take the first step towards establishing a theory of Parker vectors for infinite permutation groups, with an emphasis towards oligomorphic groups. We show that, on the one hand, many results for finite groups extend naturally to the infinite case (Parker’s Lemma, multiplicative properties, etc.), while on the other, in the infinite case some genuinely new phenomena arise. We also note that calculating Parker vectors of oligomorphic groups is akin to counting circulant combinatorial objects, mirroring in a sense the combinatorial meaning of the orbit-counting sequence of an oligomorphic group. Finally we explicitly find the Parker vectors for some groups, one of which being the automorphism group of the Rado graph.