On the set of discrete subgroups of bounded covolume in a semisimple group

In this noteG is a locally compact group which is the product of finitely many groups G_s(k_s)(s∈S), where k_s is a local field of characteristic zero and G_s an absolutely almost simplek _s-group, ofk _s-rank ≥1. We assume that the sum of the r_s is ≥2 and fix a Haar measure onG. Then, given a constantc > 0, it is shown that, up to conjugacy,G contains only finitely many irreducible discrete subgroupsL of covolume ≥c (4.2). This generalizes a theorem of H C Wang for real groups. His argument extends to the present case, once it is shown thatL is finitely presented (2.4) and locally rigid (3.2).     

On complemented subgroups of finite groups

A subgroup H of a group G is said to be complemented in G if there exists a subgroup K of G such that G = HK and H ⋂ K = 1. In this paper we determine the structure of finite groups with some complemented primary subgroups, and obtain some new results about p-nilpotent groups.     

On the prefrattini subgroups of finite soluble groups

We study the partially prefrattini groups of a finite soluble group. We prove that the set of all partially prefrattini subgroups associated with the Gaschütz system of complements to crowns is a Boolean lattice.     

Basic subgroups in abelian group rings

Suppose is a commutative ring with identity of prime characteristic and is an arbitrary abelian -group. In the present paper, a basic subgroup and a lower basic subgroup of the -component and of the factor-group of the unit group in the modular group algebra are established, in the case when is weakly perfect. Moreover, a lower basic subgroup and a basic subgroup of the normed -component and of the quotient group are given when is perfect and is arbitrary whose is -divisible. These results extend and generalize a result due to Nachev (1996) published in Houston J. Math., when the ring is perfect and is -primary. Some other applications in this direction are also obtained for the direct factor problem and for a kind of an arbitrary basic subgroup.     

The number of chains of subgroups of a finite cyclic group

T?rn?uceanu and Bentea [M. T?rn?uceanu, L. Bentea, On the number of fuzzy subgroups of finite abelian groups, Fuzzy Sets and Systems 159 (2008) 1084-1096] gave an explicit formula for the number of chains of subgroups in the lattice of a finite cyclic group by finding its generating function of one variable. Using this result T?rn?uceanu [M. T?rn?uceanu, Fuzzy subgroups of finite cyclic groups and Delannoy numbers, European J. Combin. 30 (2009) 283-287] found an explicit formula for the central Delannoy number. In this note we find a generating function of multi-variables for the number of chains of subgroups in the lattice of subgroups of a finite cyclic group. As results we give simplified formulas for the number of chains of subgroups in the lattice of subgroups of a finite cyclic group and for the central Delannoy numbers compared with the formulas given by T?rn?uceanu and Bentea.     

The order of elements in Sylow p-subgroups of the symmetric group

Define a random variable ξ_n by choosing a conjugacy class C of the Sylow p-subgroup of S_pn by random, and let ξ_n be the logarithm of the order of an element in C. We show that ξ_n has bounded variance and mean order log n /log p +O(1), which differs greatly from the average order of elements chosen with equal probability. This revised version was published online in June 2006 with corrections to the Cover Date.     

The average number of Sylow subgroups of a finite group

We prove that if the average Sylow number (ignoring the Sylow numbers that are one) of a finite group G is ?7, then G is solvable.     

Vector invariants for the two-dimensional modular representation of a cyclic group of prime order

In this paper, we study the vector invariants of the 2-dimensional indecomposable representation V₂ of the cyclic group, Cp, of order p over a field F of characteristic p, FCp[mV₂]. This ring of invariants was first studied by David Richman (1990) [20] who showed that the ring required a generator of degree m(p−1), thus demonstrating that the result of Noether in characteristic 0 (that the ring of invariants of a finite group is always generated in degrees less than or equal to the order of the group) does not extend to the modular case. He also conjectured that a certain set of invariants was a generating set with a proof in the case p=2. This conjecture was proved by Campbell and Hughes (1997) in [3]. Later, Shank and Wehlau (2002) in [24] determined which elements in Richman's generating set were redundant thereby producing a minimal generating set.We give a new proof of the result of Campbell and Hughes, Shank and Wehlau giving a minimal algebra generating set for the ring of invariants FCp[mV₂]. In fact, our proof does much more. We show that our minimal generating set is also a SAGBI basis for FCp[mV₂]. Further, our results provide a procedure for finding an explicit decomposition of F[mV₂] into a direct sum of indecomposable Cp-modules. Finally, noting that our representation of Cp on V₂ is as the p-Sylow subgroup of SL₂(Fp), we describe a generating set for the ring of invariants F[mV₂]^SL₂(Fp) and show that (p+m−2)(p−1) is an upper bound for the Noether number, for m>2.     

On the existence of f-local subgroups in a group

We prove the existence of infinite subgroups with nontrivial locally finite radicals and of locally finite subgroups in the groups with almost finite almost solvable elements of prime orders and in the groups with generally finite elements.     

The spectrum of the Burnside Tambara functor of a cyclic group

We derive a family of prime ideals of the Burnside Tambara functor for a finite group G. In the case of cyclic groups, this family comprises the entire prime spectrum. We include some partial results towards the same result for a larger class of groups.     

Some sufficient conditions on the number of non-abelian subgroups of a finite group to be solvable

Thompson’s theorem indicates that a finite group with a nilpotent maximal subgroup of odd order is solvable. As an important application of Thompson’s theorem, a finite group is solvable if it has an abelian maximal subgroup. In this paper, we give some sufficient conditions on the number of non-abelian subgroups of a finite group to be solvable.     

On computing the number of subgroups of a finite Abelian group

We consider the numberN _A (r) of subgroups of orderp ^r ofA, whereA is a finite Abelianp-group of type =₁,₂,..., _l ()), i.e. the direct sum of cyclic groups of order ⁱⁱ. Formulas for computingN _A (r) are well known. Here we derive a recurrence relation forN _A (r), which enables us to prove a conjecture of P. E. Dyubyuk about congruences betweenN _A (r) and the Gaussian binomial coefficient .     

On the number of maximal subgroups of a finite solvable group

For a finite solvable group G and prime number p, we use elementary methods to obtain an upper bound for \mathfrak m_p(G){\mathfrak {m}_{p}(G)} , defined as the number of maximal subgroups of G whose index in G is a power of p. From this we derive an upper bound on the total number of maximal subgroups of a finite solvable group in terms of its order. This bound improves existing bounds, and we identify conditions on the order of a finite solvable group under which this bound is best possible.     

有限群的ss-置换子群

设G为有限群,称G的子群H为ss-置换子群,如果存在G的次正规子群B使得G=HB,且H与B的任意Sylow子群可以交换,即对任意X∈Syl（B）有XH=HX.利用子群的ss-置换性来研究有限群的结构,得到有限群超可解的两个充分条件.     

Basic subgroups in modular abelian group algebras

Suppose F is a perfect field of char F = p ≠ 0 and G is an arbitrary abelian multiplicative group with a p-basic subgroup B and p-component G _p . Let FG be the group algebra with normed group of all units V(FG) and its Sylow p-subgroup S(FG), and let I _p (FG; B) be the nilradical of the relative augmentation ideal I(FG; B) of FG with respect to B. The main results that motivate this article are that 1 + I _p (FG; B) is basic in S(FG), and B(1 + I _p (FG; B)) is p-basic in V(FG) provided G is p-mixed. These achievements extend in some way a result of N. Nachev (1996) in Houston J. Math. when G is p-primary. Thus the problem of obtaining a (p-)basic subgroup in FG is completely resolved provided that the field F is perfect. Moreover, it is shown that G _p (1 + I _p (FG; B))/G _p is basic in S(FG)/G _p , and G(1 + I _p (FG; B))/G is basic in V(FG)/G provided G is p-mixed. As consequences, S(FG) and S(FG)/G _p are both starred or divisible groups. All of the listed assertions enlarge in a new aspect affirmations established by us in Czechoslovak Math. J. (2002), Math. Bohemica (2004) and Math. Slovaca (2005) as well.     

On maximal subgroups of the multiplicative group of a division algebra

The question of existence of a maximal subgroup in the multiplicative group D^* of a division algebra D finite-dimensional over its center F is investigated. We prove that if D^* has no maximal subgroup, then deg(D) is not a power of 2, F^*2 is divisible, and for each odd prime p dividing deg(D), there exist noncyclic division algebras of degree p over F.