Augmentation quotients and dimension subgroups of semidirect products

Cohomology of Semidirect Product Groups

The isomorphism problem for finite groups of order $n$ (GpI) has long been known to be solvable in $n^{\log n+O(1)}$ time, but only recently were polynomial-time algorithms designed for several interesting group classes. Inspired by recent progress, we revisit the strategy for GpI via the extension theory of groups. The extension theory describes how a normal subgroup $N$ is related to $G/N$ via $G$, and this naturally leads to a divide-and-conquer strategy that “splits” GpI into two subproblems: one regarding group actions on other groups, and one regarding group cohomology. When the normal subgroup $N$ is abelian, this strategy is well known. Our first contribution is to extend this strategy to handle the case when $N$ is not necessarily abelian. This allows us to provide a unified explanation of all recent polynomial-time algorithms for special group classes. Guided by this strategy, to make further progress on GpI, we consider central-radical groups, proposed in Babai et al. [Code equivalence and group isomorphism, in Proceedings of the 22nd Annual ACM--SIAM Symposium on Discrete Algorithms (SODA'11), SIAM, Philadelphia, 2011, ACM, New York, pp. 1395--1408]: the class of groups such that $G$ modulo its center has no abelian normal subgroups. This class is a natural extension of the group class considered by Babai et al. [Polynomial-time isomorphism test for groups with no abelian normal subgroups (extended abstract), in International Colloquium on Automata, Languages, and Programming (ICALP), 2012, pp. 51--62], namely those groups with no abelian normal subgroups. Following the above strategy, we solve GpI in $n^{O(\log \log n)}$ time for central-radical groups, and in polynomial time for several prominent subclasses of central-radical groups. We also solve GpI in $n^{O(\log\log n)}$ time for groups whose solvable normal subgroups are elementary abelian but not necessarily central. As far as we are aware, this is the first time there have been worst-case guarantees on an $n^{o(\log n)}$-time algorithm that tackles both aspects of GpI---actions and cohomology---simultaneously. Prior to this work, the best proven upper bounds on algorithms for groups with central radicals were $n^{O(\log n)}$, even for groups with a central radical of constant size, such as ${Rad}(G) = Z(G)=\mathbb{Z}_2$. To develop our new algorithms we utilize several mathematical results on the detailed structure of cohomology classes, as well as algorithmic results for code equivalence, coset intersection, and cyclicity testing of modules over finite-dimensional associative algebras. We also suggest several promising directions for future work.

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._

Dimension subgroups of semi-direct products

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.

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).

Extremal and maximal normal abelian subgroups of a maximal unipotent subgroup in groups of Lie type

The isomorphism problem for groups given by their multiplication tables (GPI) has long been known to be solvable in n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O(log n)</sup> time, but only recently has there been significant progress towards polynomial time. For example, Babai et al. (ICALP 2012) gave a polynomial-time algorithm for groups with no abelian normal subgroups. Thus, at present it is crucial to understand groups with abelian normal subgroups to develop n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">o(log n)</sup> -time algorithms. Towards this goal we advocate a strategy via the extension theory of groups, which describes how a normal subgroup N is related to G/N via G. This strategy "splits" GPI into two sub problems: one regarding group actions on other groups, and one regarding group cohomology. The solution of these problems is essentially necessary and sufficient to solve GPI. Most previous works naturally align with this strategy, and it thus helps explain in a unified way the recent polynomial-time algorithms for other group classes. In particular, most prior results in the multiplication table model focus on the group action aspect, despite the general necessity of cohomology, for example for p-groups of class 2-believed to be the hardest case of GPI. To make progress on the group cohomology aspect of GPI, we consider central-radical groups, proposed in Babai et al. (SODA 2011): the class of groups such that G mod its center has no abelian normal subgroups. Recall that Babai et al. (ICALP 2012) consider the class of groups G such that G itself has no abelian normal subgroups. Following the above strategy, we solve GPI in n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O(log log n)</sup> time for central-radical groups, and in polynomial time for several prominent subclasses of central-radical groups. We also solve GPI in n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O(log log n)</sup> -time for groups whose solvable normal subgroups are elementary abelian but not necessarily central. As far as we are aware, this is the first time that a nontrivial algorithm with worst-case guarantees has tackled both aspects of GPI-actions and cohomology-simultaneously. Prior to this work, only n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O(log n)</sup> -time algorithms were known, even for groups with a central radical of constant size, such as Z(G) = Z_2. To develop these algorithms we utilize several mathematical results on the detailed structure of cohomology classes, as well as algorithmic results for code equivalence, coset intersection and cyclicity testing of modules over finite-dimensional associative algebras. We also suggest several promising directions for future work.

A group is said to be hyperabelian if each of its non-trivial quotient groups has a non-trivial abelian normal subgroup, and subsoluble if each of its non-trivial quotient groups has a non-trivial abelian subnormal subgroup. In this note we settle a point raised by Robinson ([2], p. 87) by showing that subsoluble groups satisfying Min-n, the minimal condition for normal subgroups, need not be hyperabelian. More exactly, we construct a group G whose normal subgroups are well-ordered by inclusion, of order-type ω + 1, having a perfect minimal normal subgroup N which is generated by its abelian normal subgroups, such that G/N is locally soluble and hyperabelian; G is obviously a group satisfying Min-n which is subsoluble but not hyperabelian. Our construction uses the notion of the treble product rower of a family of groups introduced in [1].

Let G be an arbitrary group and Zn(G) denote the group algebra of G over the integers modulo n. If δi(G) denotes ith power of the augmentation ideal δ(G) of Zn(G), thenis easily seen to be a normal subgroup of G. It is denoted by Di, n(G) and is called ith dimension subgroup of G modulo n. It can be shown that these dimension subgroups are determined by the dimension subgroups modulo a power of a prime p. Hence we shall restrict our attention to these dimension subgroups.

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.

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' ⊕