On Fox and augmentation quotients of semidirect products

Cohomology of Semidirect Product Groups

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.

A theorem of a rather general nature is proved, which gives a positive solution to the restricted Burnside problem for a variety of groups with operators whose identities are obtained by “operator diluting” (in some precise sense) ordinary identities defining a variety of groups for which this problem has a positive solution. Namely, let be a finite group, a family of -operator identities, and a family of (ordinary) group identities obtained from by replacing all operators by 1. Suppose that the associated Lie ring of a free group in the variety defined by satisfies a system of multilinear identities that defines a locally nilpotent variety of Lie rings with a function bounding the nilpotency class of a -generator Lie ring in this variety. It is proved that if, for a -generator -group , the semidirect product is nilpotent, then the nilpotency class of is at most . A strong condition that be nilpotent is automatically satisfied if both and are finite -groups. Instead of the condition on the identities of the associated Lie ring, an analogous condition on the identities could be required, but such a condition would be stronger. An example at the end of the paper shows that the word multilinear in this condition is essential. It is not yet clear whether the condition that be finite is essential, and whether one can choose a function from the conclusion to be independent of . Earlier, in [1], a similar theorem on nilpotency in varieties of groups with operators was proved by the author. The author's results on groups with splitting automorphisms of prime order (see [2], [3]) are prototypes for both papers on operator groups. Bibliography: 18 titles.

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

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._

Let D = FlxF2x- xF" be a direct product of n free groups Ft, F2, , Fn, a an automorphism of D which leaves all but one of the noncyclic factors in D pointwise fixed and T an infinite cyclic group.Let D x T be the semidirect product of D and T with respect to a.We prove that the Whitehead group of D x T and the projective class group of the integral group ring Z(D x T) are trivial.The second result implies that the projective class group of the integral group ring over the fundamental group of a surface is trivial.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 K0Z(G).Let a be an automorphism of G and let F be an infinite cyclic group.Then we denote by G x x F the semidirect product of G and F with respect to a.Let M be a connected 2-dimensional manifold and trx(M) the fundamental group of M. If M is open, then ttx(M) is a free group so that K0Z(ttx(M)) is trivial by a theorem of Bass (cf.[I]).Next, if Misasphereor a projective plane, then trx(M)=0 or T2 (cyclic group of order 2) and so K0Z(ttx(Mj) =0 (cf.[7, p. 419]).Now, if Mis closed and is not a sphere or projective plane, then Farrell-Hsiang[4] have shown that trx(M) is just the semidirect product F xxT, where F is a free group.The purpose of this paper is to show that K0Z(F xx T)=0 and so the projective class group of the integral group ring over the fundamental group of a surface is always trivial.In fact, we prove:

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.

Let G be a group, ZG the integral group ring of G and Δ(G) its augmentation ideal. M. Khambadkone [2] studies the quotient group Δ(G) Δ(H)/Δ 2 (G) Δ(H) when is a normal subgroup of G and gives the result that if G is the semidirect product H |>K of a finitely generated normal subgroup H by a subgroup K, then Δ(G) Δ(H)/Δ 2 (G) Δ(H) ≅ K|K'⊗ HIH' ⊕

Let H be a generalized dihedral, semi-dihedral, quaternion, or modular group, and let A = (u, v, w) be a product of three odd order cyclic groups, with (|v|,|w|) = 1. For R a semi-local Dedekind domain of characteristic 0 in which no prime divisor of |H|.|A| is invertible, we prove that there is a semi-direct product G = H × A such that the group ring RG has an exceptional automorphism, i.e. provides a counter-example to a well-known conjecture of Zassenhaus on automorphisms of group rings

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.

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