Non-normal Subgroups Research Articles

On (2 n,2,2 n, n) relative difference sets

In this article, we show that a (2 n,2,2 n, n) relative difference set in a group G of order 4 n exists only if a Sylow 2-subgroup of G is non-cyclic and n is even unless n=1. We also construct (2 n,2,2 n, n) relative difference sets relative to non-normal subgroups.

Exponential nonlocal symmetries and nonnormal reduction of order

The conventional approach to double reduction of the order of an ordinary differential equation using Lie symmetries is via the normal subgroups of point symmetries. We show that, provided that one is prepared to use nonlocal symmetries, initial reduction by the nonnormal subgroup does not prevent the double reduction. We further illustrate our results with the general third-order equations invariant under the nonsolvable algebras, sl(2, R) (of which the Chazy equation is a noted example) and so(3).

Groups whose non-normal subgroups have a transitive normality relation

This article investigates the structure of groups in which every subgroup either is subnormal or has a transitive normality relation, with special attention to the case in which subnormal subgroups have bounded defect.

Groups with Polycyclic Non-Normal Subgroups

The structure of groups in which many subgroups have a certain property X has been investigated for several choices of the property X. Groups whose non-normal subgroups satisfy certain finite rank conditions are studied in this article. In particular, a classification of groups in which every subgroup is either normal or polycyclic is given.

Groups with Polycyclic Non-Normal Subgroups

The structure of groups in which many subgroups have a certain property X has been investigated for several choices of the property X. Groups whose non-normal subgroups satisfy certain finite rank conditions are studied in this article. In particular, a classification of groups in which every subgroup is either normal or polycyclic is given.

Groups with a Bounded Number of Conjugacy Classes of Non-normal Subgroups

Let ν(G) denote the number of conjugacy classes of non-normal subgroups of a groupG. We prove that ifGis a finite group and ν(G)≠0, then there is a cyclic subgroupCof prime power order contained in the centre ofGsuch that the order ofG/Cis a product of at most ν(G)+1 primes. We also obtain a bound in the opposite direction, thus obtaining a criterion for a group to have a bounded number of conjugacy classes of non-normal subgroups. These results extend to infinite groups, with the subgroupCbeing the infinite Prüferp-group, but only whenGhas finitely many non-normal subgroups. This is to be expected because of the existence of monsters of the type constructed by S. V. Ivanov and A. Yu. Ol'shanskii. The structure of groups with infinitely many non-normal subgroups falling into finitely many conjugacy classes is also studied.

Morita Equivalent Blocks in Non-Normal Subgroups and p -Radical Blocks in Finite Groups

Let O be a complete discrete valuation ring with unique maximal ideal J(O), let K be its quotient field of characteristic 0, and let k be its residue field O/J(O) of prime characteristic p. We fix a finite group G, and we assume that K is big enough for G, that is, K contains all the ∣G∣-th roots of unity, where ∣G∣ is the order of G. In particular, K and k are both splitting fields for all subgroups of G. Suppose that H is an arbitrary subgroup of G. Consider blocks (block ideals) A and B of the group algebras RG and RH, respectively, where R∈{O, k}. We consider the following question: when are A and B Morita equivalent? Actually, we deal with ‘naturally Morita equivalent blocks A and B’, which means that A is isomorphic to a full matrix algebra of B, as studied by B. Külshammer. However, Külshammer assumes that H is normal in G, and we do not make this assumption, so we get generalisations of the results of Külshammer. Moreover, in the case H is normal in G, we get the same results as Külshammer; however, he uses the results of E. C. Dade, and we do not.

On finite groups with few non-normal subgroups

In this paper we characterize the finite groups having exactly two conjugacy classes of non-normal subgroups.

On the normalizer property for integral group rings

Let R(G) denote the intersection of all nonnormal subgroups of a group G. In this note, we prove that for every finite group G, if R(G) is not trivial, then the normalizer property holds forG.

Some explicit bounds in groups with a finite number of non-normal subgroups

The number of conjugacy classes of non-normal subgroups is an invariant of a group G denoted by v{G). In this paper explicit upper bounds for the order of the commutator subgroup G' and for the index of the centre in G, [G : Z(G)] are given for a group G with only a finite number of non-normal subgroups. The bounds provided are functions of v(G) and the primes appearing as orders of elements of G.

Finite equilibrated groups

If H, K are subgroups of a group G, then HK is a subgroup of G if and only if HK = KH. This condition certainly holds if H ≤ NG(K) or K ≤ NG(H). But the majority of groups can also be expressed as HK, where neither H nor K is normal. In this paper we consider groups G for which no subgroup G1 can be expressed as the product of non-normal subgroups of G1. Such a group is said to be equilibrated. Thus G is equilibrated if and only if either H ≤ NG(K) or K ≤ NG(H) whenever H, K and HK are subgroups of G.

Groups with Anomalous Automorphisms

Let Autnn(G) be the group of all automorphisms of a group G that fix all nonnormal subgroups of G. It is shown that Autnn(G) has a very restricted structure.

The number of conjugacy classes of non-normal subgroups in nilpotent groups

In a recent paper, Rolf Brandi classified all finite groups having exactly one conjugacy class of nonnormal subgroups, and conjectured thatfor a nilpotent group G of nilpotency class c = c(G) the number v(G) = vof conjugacy classes of nonnormal subgroups satisfies the inequality v(G) ≥ c(G) – 1 (with the exception of the Hamiltonian groups, of course). The purpose of this paper is to establish this conjecture and to decide when this inequality is sharp.

On automorphisms fixing non-normal subgroups of groups

On automorphisms fixing non-normal subgroups of groups

Positive definite functions and relative property (T) for subgroups of discrete groups

Relative Property (T) for a subgroup H of a group G and its connection with positive definite functions are studied. A relation with the Haagerup approximation property is established. We show that if H is a non-normal subgroup of a group G which has Property (T) and G/H is amenable as a graph then H has finite index in G.

Groups with a maximality condition for some non-normal subgroups

We describe (generalized) soluble-by-finite groups in which the set of non-normal subgroups which are not finitely generated satisfies the maximal condition.

Groups with few non-normal subgroups

Finite groups are characterized all of whose non-normal subgroups are conjugate

Smash products for {$G$}-sets, Clifford theory and duality theorems

The categorical approach to the theory of group graded rings has been successful in recent years and in particular the use of smash products has provided several new ideas, cf. [4] and later [3], [10], [11], ... Another tool, the use of which is prompted upon us by the problems connected to the consideration of induced modules for a Ggraded ring R with respect to a non-normal subgroup H of G, is the G-set theory, cf. [11]. The latter considerations stem from a general Clifford theory for group graded rings initiated by E. Dade, cf. [6] and [7] and extended recently in [9] and [12] to almost complete generality as far as graded modules are being concerned. However the G-set graded modules remained to be studied and in view of the Clifford theory with respect to a non-normal subgroup and the interest of the generalized Hecke algebras appearing in this theory, cf. [12], it is worthwhile to develop in some detail a theory of smash products for G-sets and its use in Clifford theory and duality theorems as in [4], [2].

Hidden symmetries and linearization of the modified Painlevé–Ince equation

The linearization and hidden symmetries of the modified Painlevé–Ince equation, y″+σyy′+βy3=0, where σ and β are constants, are presented. The linearization of this equation by a nonlocal transformation yields a damped (stable) or growing (unstable) harmonic oscillator equation for β≳0. Hidden symmetries are analyzed by transforming the modified Painlevé–Ince equation to a third-order ordinary differential equation (ODE) which, in general, is invariant under a three-parameter group by a Riccati transformation. A type I hidden symmetry is found of a second-order ODE found from the third-order ODE where a symmetry is lost in the reduction of order by the non-normal subgroup invariants. A type II hidden symmetry occurs in the third-order ODE because the symmetries of a second-order ODE, reduced from the third-order ODE by another set of normal subgroup invariants, are increased.

On a hyperbolic 3-manifold with some special properties

We present a closed hyperbolic 3-manifold M with some surprising properties. The universal covering group of M is a normal torsion-free subgroup of minimal index in one of the nine Coxeter groups G, generated by the reflections in the faces of one of the nine Lannér-tetrahedra (bounded tetrahedra in hyperbolic 3-space all of whose dihedral angles are of the form π/n with n ∈ ℕ see [1] or [3]). The corresponding Coxeter group G splits as a semidirect product G = π1M⋉A, where A is a finite subgroup of G, and G is the only one of the nine Coxeter groups associated to the Lannér-tetrahedra which admits such a splitting (this follows using results in [4]). We derive a presentation of π1M and show that the first homology group H1(M) of M is isomorphic to ℚ11. This is in sharp contrast to other torsion-free (non-normal) subgroups of finite index in Coxeter groups constructed in [1] which all have finite first homology (though it is known that they are all virtually ℚ-representable (see [5], p. 434). It follows from our computations that the Heegaard genus of M is 11, and that there exists a Heegaard splitting of M of genus 11 invariant under the action of the group I+(M) ≌ S5 ⊕ ℚ2 of orientation-preserving isometries of M (we compute this group in [4]), so that the Heegaard genus of M is equal to the equivariant Heegaard genus of the action of I+(M) on M. Moreover M is maximally symmetric in the sense of [4, 6]: the order 120 of the subgroup of index 2 in I+(M) which preserves both handle-bodies of the Heegaard splitting is the maximal possible order of a group of orientation-preserving diffeomorphisms of a handle-body of genus 11. (This maximal order is 12(g—1) for a handle-body of genus g; see [7].) By taking the coverings Mq of M corresponding to the surjections π1M→H1(M) ≌ ℚ11→(ℚq)11 for q ∈ ℕ, we obtain explicitly an infinite series of maximally symmetric hyperbolic 3-manifolds.