Conjugacy Classes Of Non-normal Subgroups Research Articles

On the Conjugacy Classes of Cyclic Non-normal Subgroups

Let G be a finite p-group. Assume that $$\nu (G)$$ and $$\nu _c(G)$$ denote the number of conjugacy classes of non-normal subgroups and non-normal cyclic subgroups of G, respectively. In this paper, we completely classify the finite p-groups with $$\nu _c=p$$ or $$p+1$$ for an odd prime number p. Also, we classify the groups G with $$\nu (G)=\nu _c(G)=p^i, i\ge 1$$ .

The number of conjugacy classes of nonnormal subgroups of finite p-groups (III)

Assume [Formula: see text] is a positive integer and [Formula: see text] is an odd prime. Let [Formula: see text] be the number of conjugacy classes of nonnormal subgroups of a finite group [Formula: see text] and [Formula: see text] is a finite [Formula: see text]-group[Formula: see text]. The set [Formula: see text] has been determined for [Formula: see text]. In this paper, the set [Formula: see text] is determined, and it is discovered that there are three gaps in the values that [Formula: see text] can take in the case of finite [Formula: see text]-groups.

The number of conjugacy classes of nonnormal subgroups of finite p-groups (II)

Assume k is a positive integer and p is an odd prime. Let be the number of conjugacy classes of nonnormal subgroups of a finite group G and is a finite p-group The set have been determined for (see [8]). In this article, the set is determined, and it is discovered that there are two new gaps in the values that can take in the case of finite p-groups.

The number of conjugacy classes of nonnormal subgroups of finite p-groups

Assume k is a positive integer and p is a prime. Let ν(G) be the number of conjugacy classes of nonnormal subgroups of a finite group G and NCN(p,k)={ν(G)∈[0,kp]|G is a finite p-group}. In this paper, the set NCN(p,k) is determined for k≤2, and it is discovered that there is a new gap in the values that ν(G) can take in the case of finite p-groups. In particular, the number of conjugacy classes of nonnormal subgroups of minimal non-abelian p-groups is determined.

On Solvability of Finite Groups with Few Non-Normal Subgroups

For a finite group G, let v(G) denote the number of conjugacy classes of non-normal subgroups of G and vc(G) denote the number of conjugacy classes of non-normal noncyclic subgroups of G. In this paper, we show that every finite group G satisfying v(G) ≤2|π(G)| or vc(G) ≤ |π(G)| is solvable, and for a finite nonsolvable group G, v(G) = 2|π(G)| +1 if and only if G ≅ A 5.

Non-nilpotent groups with three conjugacy classes of non-normal subgroups

‎For a finite group $G$ let $nu(G)$ denote the number of conjugacy classes of non-normal subgroups of $G$‎. ‎The aim of this paper is to classify all the non-nilpotent groups with $nu(G)=3$‎.

Finite Nilpotent Groups Having Exactly Four Conjugacy Classes of Non-normal Subgroups

Finite nilpotent groups having exactly four conjugacy classes of non-normal subgroups are classified.

Groups with finitely many conjugacy classes of non-normal subgroups of infinite rank

It is proved that if a locally soluble group of infinite rank has only finitely many non-trivial conjugacy classes of subgroups of infinite rank, then all its subgroups are normal.

Conjugacy classes of non-normal subgroups of finite p-groups

Let G be a finite p-group, and let ν(G) denote the number of conjugacy classes of non-normal subgroups of G. It is known that either ν(G) ≤ 1 or ν(G) ≥ p. We determine all p-groups G with ν(G) ≤ p + 1.

Non-soluble groups with few conjugacy classes of non-normal subgroups

For a finite group G, let ν(G) denote the number of conjugacy classes of non-normal subgroups of G. We determine all finite non-soluble groups G with ν(G) < 14. As a consequence, all finite groups G with ν(G) ≤ 6 or \({\nu(G)\in \{ 8,11,12\}}\) are soluble.

The finite p-groups with p conjugacy classes of non-normal subgroups

Let ν(G) be the number of conjugacy classes of non-normal subgroups of a finite group G. The finite groups for which ν(G) ≤ 2 were determined by Dedekind and by Schmidt in the early times of group theory. On the other hand, if G is a finite p-group, La Haye and Rhemtulla have proved that either ν(G) ≤ 1 or ν(G) ≥ p. In this note, we determine all finite p-groups satisfying ν(G) = p for p > 2.

Bounds for the Number of Conjugacy Classes of Non-Normal Subgroups in a Finite p-Group

Let ν(G) be the number of conjugacy classes of non-normal subgroups of a finite group G. We obtain two new lower bounds for ν(G) when G is a non-abelian finite p-group and p is odd. More precisely, if |G| =p n , exp Z(G) = p e , and exp G/G′ =p f , let us define λ(G) = n − e and κ(G) = n − f. Then we prove that ν(G) ≥ p(λ(G) −3) +2 and ν(G) ≥ p(κ(G) −3) +2. The first bound improves the bound ν(G) ≥ λ(G) −1 given by [10], and almost in every case, the second one improves the bound ν(G) ≥ p(k − 1) +1 obtained by [6], where k is defined by the condition that |G′| =p k .

Groups with few conjugacy classes of non-normal subgroups

Abstract The structure of groups with finitely many non-normal subgroups is well known. In this paper, groups are investigated with finitely many conjugacy classes of non-normal subgroups with a given property. In particular, it is proved that a locally soluble group with finitely many non-trivial conjugacy classes of non-abelian subgroups has finite commutator subgroup. This result generalizes a theorem by Romalis and Sesekin on groups in which every non-abelian subgroup is normal.

Conjugacy classes of non-normal subgroups in finite nilpotent groups

Let v(G) be the number of conjugacy classes of non-normal subgroups of a finite group G. Poland and Rhemtulla [J. Poland and A. Rhemtulla. The number of conjugacy classes of non-normal subgroups in nilpotent groups. Comm. Algebra24 (1996), 3237–3245.] proved that if G is nilpotent of class c then c − 1 unless G is a Hamiltonian group. The sharpness of this lower bound is a problem about finite p-groups, since , where μ(G) denotes the number of normal subgroups of G. In this paper, we show that the bound c − 1 can be substantially improved: if G is a finite p-group and , then p(k − 1) + 1, unless G is a Hamiltonian group or a generalized quaternion group. In these exceptional cases, G is a 2-group and 2(k − 1).

On the number of conjugacy classes of normalisers in a finite p-group

In 1996 Poland and Rhemtulla proved that the number v (G) of conjugacy classes of non-normal subgroups of a non-Hamiltonian nilpotent group G is at least c − 1, where c is the nilpotency class of G. In this paper we consider the map that associates to every conjugacy class of subgroups of a finite p-group the conjugacy class of the normaliser of any of its representatives. In spite of the fact that this map need not be injective, we prove that, for p odd, the number of conjugacy classes of normalisers in a finite p-group is at least c (taking into account the normaliser of the normal subgroups). In the case of p-groups of maximal class we can find a better lower bound that depends also on the prime p.

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.

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.

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.

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.