Augmentation Quotients of Integral Group Rings

Cohomology of Semidirect Product Groups

Let G be an analytic group.Let Q(G) be the union of all compact subgroups of G .We give a necessary and sufficient condition for 2(G) to be dense in G in terms of the action of a maximal compact torus T of G on the nilradical TV of G.Let F be a locally compact group.Let Q(F) be the union of all compact subgroups of F .We study the problem: when Q(F) is dense in F. If F is not connected, the problem is too broad to have any meaningful answers.On the other hand, if F is almost connected, i.e., F Fo is compact where Fo is the identity component of F , then the problem is quickly reduced to the case where F is a Lie group with finitely many components.This is so because an almost connected locally compact F has a maximal compact normal subgroup M so that F M is a Lie group with finitely many components.It is easy to see that 2(F) is dense in F if and only if Q(F/A/) is dense in F/M.Let G = F M. Let C70 be the identity component of G. Since the identity component Go of G is an open subgroup, so Q(C7) n Go is dense in Go when Ci(G) is dense in G (the converse is also true, cf.Theorem 2.10).Therefore, for most of this note we shall assume that G is an analytic group.Now, let G be an analytic group with Q(G) dense in G. Let M be the maximal compact normal subgroup of G. Again, Ci(G) is dense in G if and only if Q(G/M) is dense in G/M, so we may assume that M is trivial.Let A be the nilradical of G, i.e., the maximal analytic nilpotent normal subgroup of G. Then N is simply connected since M is trivial.Furthermore, by an argument due to Djokovic [1] we can show that A is uniform in G.This implies that G is a semidirect product A K with K a compact analytic group.Hence K acts on A as a group of automorphisms.The purpose of the present note is to show the following statement.Theorem 2.7.Let G be a semidirect product N K with A a simply connected analytic nilpotent group and K a compact analytic group.Let T be a maximal torus of K. Then Q(t7) is dense in G if and only if the only element in N fixed by T is the identity element.Another characterization of Q(C7) being dense in G is the following condition._

A normal Hall subgroup $N$ of a group $G$ is a normal subgroup with its order coprime with its index. Schur-Zassenhaus theorem states that every normal Hall subgroup has a complement subgroup, that is a set of coset representatives H which also forms a subgroup of G. In this paper, we present a framework to test isomorphism of groups with at least one normal Hall subgroup, when groups are given as multiplication tables. To establish the framework, we first observe that a proof of Schur-Zassenhaus theorem is constructive, and formulate a necessary and sufficient condition for testing isomorphism in terms of the associated actions of the semidirect products, and isomorphisms of the normal parts and complement parts. We then focus on the case when the normal subgroup is abelian. Utilizing basic facts of representation theory of finite groups and a technique by Le Gall [STACS 2009], we first get an efficient isomorphism testing algorithm when the complement has bounded number of generators. For the case when the complement subgroup is elementary abelian, which does not necessarily have bounded number of generators, we obtain a polynomial time isomorphism testing algorithm by reducing to generalized code isomorphism problem. A solution to the latter can be obtained by a mild extension of the singly exponential (in the number of coordinates) time algorithm for code isomorphism problem developed recently by Babai. Enroute to obtaining the above reduction, we study the following computational problem in representation theory of finite groups: given two representations rho and tau of a group $H$ over Z_p^d , p a prime, determine if there exists an automorphism phi:H -> H, such that the induced representation rho_phi=rho o phi and tau are equivalent, in time poly(|H|,p^d).

g*: For each normal subgroup N$ 'P(G), each reduced product of G over N is a semidirect product. (G = NB is a reduced product over a normal subgroup N by a subgroup B iff B does not contain a proper subgroup B* such that G = NB*.) F. Gross [5] has shown that for a finite solvable group G having 4!(G) = 1, splitting over each normal subgroup is sufficient for the subgroup lattice to be comple

Let H be a finite group having a fixed point free au tomorph ism c~ of order p". Consider the semidirect product G = (c~)H. It is well known that (eh) v" = 1 if h 9 H (see [3], p. 334). Put K = ( ev ) H. Then G # K and the elements in G K are p-elements. This last si tuation was considered by Kurzweil in [7]. It includes as a special case the groups having a proper generalized Hughes subgroup, i.e. those verifying G + Hr, (G) where Hp, (G) = ( x 9 G I xl" Je 1). A classical result of Hughes-Thompson and Kegel assures that if G :# H v (G) then H v (G) is ni lpotent (see [5] and [6]). Assuming that G is solvable Kurzweil showed that the Fi t t ing length of Hr, (G) (and hence that of G) is bounded by a function of n (see [7]). His bound for exceptional primes (in the Hal l -Higman sense) was improved by Har t ley and Rae as a product of their work in [4]. More recently Meixner obtained a l inear bound in [8]. Finally, in [2], the best possible bound f (Hr, (G)) < n was obtained for p odd. The case p = 2 is open. The purpose of this note is to consider the general problem. We may assume that G = ( x ) K, G K consists of p-elements and the order of x is, say, p". Assuming that G is solvable, what can be said about its Fi t t ing length? In [7] Kurzweil considered the case n = I and showed that f (K) < 2. Here we prove that f (K) < n + 1 if p is odd and the bound is best possible. The result is false for p = 2 even in the case n = 2. Our theorem is a new appl icat ion of the non-coprime Shult type theorems stated in [2]. There is another problem connected to this. Let G be a finite group having a proper subgroup H and a proper normal subgroup N of H such that H c~ H ~ < N if g 9 G H. Then G is said to be a Frobenius-Wie landt group (see [1] for more details and notation). We write (G, H, N) to indicate this situation. A theorem of Wielandt (see [1] for example) assures that, in such conditions, there exists a normal subgroup K of G such that G K = ~) (H -N) o, G = H K and H c~ K = N. Assume that H is a p-group. Then osG G K consists of p-elements. Thus we are in the above situation. Conversely, if G is p-solvable and K is a normal subgroup of G such that G K consists of p-elements then taking P 9 $1, (G) we have that (G, P, P c~ K) is an F W group. To show this observe that if x 9 G K then x acts f.p.f, on every x-invariant p '-section of K. Suppose that y 9 P c~ Po where g is a nontrivial p ' -element of G. As K is p-solvable we have a p '-section A/B of K where A and B are normal in G and g 9 A B. Then [y, g 1] 9 p c~ A < B. Thus y 9 P c~ K.

Sandling(6) determined the dimension subgroups of the semidirect product of a normal abelian subgroup and a subgroup; namely if G = NT is the semidirect product of a normal abelian subgroup N and a subgroup T, then the mth dimension subgroup Dm(G) of G is equal to [N, (m – 1) G] · Dm (T) for all m ≧ 1, where

In the classification theorems of Vinberg and Yakimova for commutative nilmanifolds, the relevant nilpotent groups have a very surprising analytic property. The manifolds are of the form \(G/K = N\rtimes K/K\) where, in all but three cases, the nilpotent group \(N\) has irreducible unitary representations whose coefficients are square integrable modulo the center \(Z\) of \(N\). Here we show that, in those three “exceptional” cases, the group \(N\) is a semidirect product \(N_{1}\rtimes \mathbb {R}\) or \(N_{1}\rtimes \mathbb {C}\) where the normal subgroup \(N_{1}\) contains the center \(Z\) of \(N\) and has irreducible unitary representations whose coefficients are square integrable modulo \(Z\). This leads directly to explicit harmonic analysis and Fourier inversion formulae for commutative nilmanifolds.

Introduced extension of a group and other concepts, several of lemmas and theorems on the construction of normal subgroups were proved to use the theory of group characters, and then shows that the construction of normal subgroups of a finite group G. Key Words: Normal Subgroup; Irreducible Character; Semi-direct Product; Regular Representation; p-Sylow Subgroup

Let the Lie group G be a semidirect product, G = SK, of a connected, closed, normal subgroup S and a closed subgroup K. Let A be a nonunitary character of S, and let KA be its stability subgroup in K. Let IA, for any irreducible representation ,u of KA, denote the representation IA of G induced by the representation Au of SKA. The representation spaces are subspaces of the distributions. We show that IAIA is ultra-irreducible when the corresponding Poisson transform is injective, and find a sufficient condition for this injectivity.

Given a Lie group $G$ and a lattice $\Gamma$ in $G$, a one-parameter subgroup $\phi$ of $G$ is said to be rigid if for any other one-parameter subgroup $\psi$, the flows induced by $\phi$ and $\psi$ on $\Gamma\backslash G$ (by right translations) are topologically orbit-equivalent only if they are affinely orbit-equivalent. It was previously known that if $G$ is a simply connected solvable Lie group such that all the eigenvalues of $\mathrm{Ad} (g) $, $g\in G$, are real, then all one-parameter subgroups of $G$ are rigid for any lattice in $G$. Here we consider a complementary case, in which the eigenvalues of $\mathrm{Ad} (g)$, $g\in G$, form the unit circle of complex numbers.Let $G$ be the semidirect product $N \rtimes M$, where $M$ and $N$ are finite-dimensional real vector spaces and where the action of $M$ on the normal subgroup $N$ is such that the center of $G$ is a lattice in $M$. We prove that there is a generic class of abelian lattices $\Gamma$ in $G$ such that any semisimple one-parameter subgroup $\phi$ (namely $\phi$ such that $\mathrm{Ad} (\phi_t)$ is diagonalizable over the complex numbers for all $t$) is rigid for $\Gamma$ (see Theorem 1.4). We also show that, on the other hand, there are fairly high-dimensional spaces of abelian lattices for which some semisimple $\phi$ are not rigid (see Corollary 4.3); further, there are non-rigid semisimple $\phi$ for which the induced flow is ergodic.

Automorphism groups of tetravalent Cayley graphs on regular [formula omitted]-groups

For an almost normal subgroup 0 of a discrete group , conditions are given which allow one to dene a universal C-norm on the Hecke algebra H( ; 0). If is a semidirect product of a normal subgroup N containing 0 by a group G satisfying some order relations arising from a naturally dened subsemigroup T , and if the normalizer of N is also normal in , then a presentation of H( ; 0) is given. In this situation the C-completion of H( ; 0 )i s-isomorphic with the semigroup crossed product C-algebra C(N= 0)oT. In their paper introducing a number theoretical model of a quantum statistical system exhibiting a phase transition with symmetry breaking, Bost and Connes introduce the notion of an almost normal subgroup 0 of a discrete group , along with the associated Hecke algebra H( ; 0) and its reduced C-algebra completion C r ( ; 0 )( [BC]). They also provide a presentation of the Hecke algebra in the context of the specic almost normal subgroup they consider in their model. A connection between these relations and some relations occurring in a stable C-algebra associated with certain examples of dynamical systems described in [B] provided the motivation for considering the Hecke algebras further. An overview of the structure of the paper follows. After some preliminaries on almost normal subgroup pairs ( ; 0) we introduce a fundamental semigroup T in the group , which contains the normalizer N 0 of 0. A basic representation of this semigroup as isometries in the convolution Hecke algebra H( ; 0) is described. In the presence of a normal subgroup N of containing 0 and contained in N 0 , a natural semigroup C-dynamical system occurs which possesses a universal property with respect to-representations of the Hecke algebra. In the second section we discuss some properties of group partial pre-order relations arising from a subsemigroup of the group in much the same spirit as Nica in [N]. Applying this to our situation, withT as the subsemigroup of , and introducing a notion of solvable least upper bounds, we obtain some conditions allowing a denition of a universal C-norm on the Hecke algebra. Assuming some more structure for the pair ( ; 0), namely that is an extension of a normal subgroup N containing 0, we obtain that

Let D=F1 x F2 x... x Fn be a direct product of n free groups F1, F2, * , F* * , ox an automorphism of D which leaves all but one of the noncyclic factors in D pointwise fixed, T an infinite cyclic group and F another free group. Let D x a T be the semidirect product of D and T with respect to a and (D x a T) x aXIdT F the semidirect product of D xa Tand F with respect to the automorphism x id T of D Xa T induced by a. We prove that the Whitehead group of (D xa, T) X 2xidT F and the projective class group of the integral group ring Z((D x a T) X aXidT F) are trivial. These results extend that of [3]. Let G be a group. We denote the Whitehead group of G by Wh G and the projective class group of the integral group ring Z(G) of G by kOZ(G). We recall the definition of semidirect product of groups and the definition of twisted group ring. For undefined terminologies used in the paper, we refer to [3] and [4]. Let oc be an automorphism of G and F a free group generated by {tA}. If w is a word in tA defining an element in F, we denote by Iwl the total exponent sum of the tA appearing in w. The semidirect product G xa F of G and F with respect to a is defined as follows: G x . F=GxF as sets and multiplication in G x . Fis given by (g, w)(g', w') = (go-lwl(g'), ww'), for any (g, w), (g', w') in G x F. In particular, if F is an infinite cyclic group T= (t) generated by t, we have the semidirect product G x a T of G and T with respect to oc. Let R be an associative ring with identity and oc an automorphism of R. Let F be a free group (or free semigroup) generated by {tA}. The otwisted group ring R,[F] of F over R is defined as follows: additively R,[F]=R[F], the group ring of F over R, so that its elements are finite linear combinations of elements in F with coefficients in R. Multiplication in R,[F] is given by (rw)(rIw')=roc-1I1(r')ww', for any rw, r'w' in R,[F]. In particular, if F is a free group (resp. free semigroup) generated by t, we Received by the editors May 25, 1973. AMS (MOS) subject classfiJcations (1970). Primary 13D15, 16A26, 18F25; Secondary 16A06, 16A54.

Any normal reflection subgroup of a Coxeter system (W, S) is a factor in a semidirect product decomposition of W as described by Bonnafé and Dyer. Namely, S is the union of two subsets I and J such that no element of I is conjugate to an element of J, is the subgroup generated by WI -conjugates of elements of J, and W is the semidirect product of WI by . This note describes the reduced expressions of elements of the form wxw–1 with w ∈ WI and x ∈ WJ in terms of reduced expressions of x and a suitable element of WI.

On Cocommutative Hopf Algebras of Finite Representation Type