Non-normal Subgroups Research Articles

On the sum of orders of non-cyclic and non-normal subgroups in a finite group

Let $G$ be a finite group and $\mathcal{C}(G)$ denote the set of all non-normal non-cyclic subgroups of $G$. In this paper, the function $\delta_c(G) =\frac{1}{|G|}\sum\limits_{H\in\mathcal{C}(G)}|H|$ is introduced. In fact, we prove that, if $\delta_c(G)\leq \frac{10}{3}$, then either $G\cong A_5$, or $G$ is solvable. We also find some examples of finite groups $G$ with $\delta_c(G)\leq \frac{10}{3}$.

New spence difference sets

AbstractSpence [9] constructed $$\left( \frac{3^{d+1}(3^{d+1}-1)}{2}, \frac{3^d(3^{d+1}+1)}{2}, \frac{3^d(3^d+1)}{2}\right) $$ 3 d + 1 ( 3 d + 1 - 1 ) 2 , 3 d ( 3 d + 1 + 1 ) 2 , 3 d ( 3 d + 1 ) 2 -difference sets in groups $$K \times C_3^{d+1}$$ K × C 3 d + 1 for d any positive integer and K any group of order $$\frac{3^{d+1}-1}{2}$$ 3 d + 1 - 1 2 . Smith and Webster [8] have exhaustively studied the $$d=1$$ d = 1 case without requiring that the group have the form listed above and found many constructions. Among these, one intriguing example constructs Spence difference sets in $$A_4 \times C_3$$ A 4 × C 3 by using (3, 3, 3, 1)-relative difference sets in a non-normal subgroup isomorphic to $$C_3^2$$ C 3 2 . Drisko [3] has a note implying that his techniques allow constructions of Spence difference sets in groups with a noncentral normal subgroup isomorphic to $$C_3^{d+1}$$ C 3 d + 1 as long as $$\frac{3^{d+1}-1}{2}$$ 3 d + 1 - 1 2 is a prime power. We generalize this result by constructing Spence difference sets in similar families of groups, but we drop the requirement that $$\frac{3^{d+1}-1}{2}$$ 3 d + 1 - 1 2 is a prime power. We conjecture that any group of order $$\frac{3^{d+1}(3^{d+1}-1)}{2}$$ 3 d + 1 ( 3 d + 1 - 1 ) 2 with a normal subgroup isomorphic to $$C_3^{d+1}$$ C 3 d + 1 will have a Spence difference set (this is analogous to Dillon’s conjecture in 2-groups, and that result was proved in Drisko’s work). Finally, we present the first known example of a Spence difference set in a group where the Sylow 3-subgroup is nonabelian and has exponent bigger than 3. This new construction, found by computing the full automorphism group $$\textrm{Aut}(\mathcal {D})$$ Aut ( D ) of a symmetric design associated to a known Spence difference set and identifying a regular subgroup of $$\textrm{Aut}(\mathcal {D})$$ Aut ( D ) , uses (3, 3, 3, 1)-relative difference sets to describe the difference set.

Groups whose non-normal subgroups are either nilpotent or minimal non-nilpotent

Groups whose non-normal subgroups are either nilpotent or minimal non-nilpotent

Finite $p$-groups in which the normalizer of each nonnormal subgroup is small

Let $G$ be a finite non-Dedekindian $p$-group which satisfies $N_G(H)=HZ(G)$ for each nonnormal subgroup $H$, we call it an $NS$-group.
 In this paper, it is proved that an $NS$-group is the product of minimal nonabelian group and the center.

Finite p-groups in which the cores of all the nonnormal subgroups are in the center

Let [Formula: see text] be a finite [Formula: see text]-group. Then [Formula: see text] is said to be a [Formula: see text]-group if [Formula: see text] for every nonnormal subgroup [Formula: see text] of [Formula: see text]. In this paper, we study the [Formula: see text]-group [Formula: see text] and get [Formula: see text]. It is proved that [Formula: see text] if [Formula: see text] and [Formula: see text] if [Formula: see text].

SOLVABLE GROUPS WHOSE NONNORMAL SUBGROUPS HAVE FEW ORDERS

Abstract Suppose that G is a finite solvable group. Let $t=n_c(G)$ denote the number of orders of nonnormal subgroups of G. We bound the derived length $dl(G)$ in terms of $n_c(G)$ . If G is a finite p-group, we show that $|G'|\leq p^{2t+1}$ and $dl(G)\leq \lceil \log _2(2t+3)\rceil $ . If G is a finite solvable nonnilpotent group, we prove that the sum of the powers of the prime divisors of $|G'|$ is less than t and that $dl(G)\leq \lfloor 2(t+1)/3\rfloor +1$ .

Finite p-groups in which the intersection of nonnormal subgroup and center has bounded order

Given a non-negative integer i, a finite p-group G is called an -group if for every nonnormal subgroup H of G. It is proved that the center of -group has order or its derived subgroup is of order . And the classification of MC(p)-groups is obtained.

A new criterion for solvability of a finite group by the sum of orders of non-normal subgroups

A new criterion for solvability of a finite group by the sum of orders of non-normal subgroups

FC-groups with Few Subnormal Non-normal Subgroups

A group G is said to be an FC-group if every conjugacy class of G has finite order and is said to be a T-group if every subnormal subgroup is normal in G. In this paper we give a characterization of groups that are both FC-groups and T-groups and go further in the study of FC-groups with few subnormal non-normal subgroups in the sense of the chain conditions. The structure of FC-groups satisfying the maximal, the minimal and the double chain condition on subnormal non-normal subgroups will be described.

The Application of GAP Software in Constructing the Non-Normal Subgroup Graphs of Alternating Groups

A graph in group theory is constructed by using any elements of a group as a set of vertices. Some of the properties of a group are used to form the edges of the graph. A finite group can be represented in a graph by its subgroup structure. A subgroup H of a group G is a subset of G, where H itself is a group under the same operation as in G, whereas a subgroup H is said to be a normal subgroup if its left and right cosets coincide. The non-normal subgroup graph of a group G is defined as a directed graph with a vertex set G and two distinct elements x and y are adjacent if xy ∈ H. In this paper, the non-normal subgroups graph of alternating groups for order twelve is determined by using GAP software. The graphs are found to be a union of complete digraphs and directed graphs with the same pattern depending on the order of the non-normal subgroups.

On non-normal cyclic subgroups of prime order or order 4 of finite groups

Abstract In this paper, we call a finite group G G an N L M NLM -group ( N C M NCM -group, respectively) if every non-normal cyclic subgroup of prime order or order 4 (prime power order, respectively) in G G is contained in a non-normal maximal subgroup of G G . Using the property of N L M NLM -groups and N C M NCM -groups, we give a new necessary and sufficient condition for G G to be a solvable T T -group (normality is a transitive relation), some sufficient conditions for G G to be supersolvable, and the classification of those groups whose all proper subgroups are N L M NLM -groups.

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 $$\nu (G)=\nu _c(G)=p^i, i\ge 1$$ .

Groups where non-normal subgroups are close to Abelian groups

In this paper we consider solvable groups, each subgroup of which is either normal or has a Chernikov commutator subgroup.

On groups, whose non-normal subgroups are either contranormal or core-free

We investigate the influence of some natural types of subgroups on the structure of groups. A subgroup H of a group G is called contranormal in G, if G = HG. A subgroup H of a group G is called core-free in G, if CoreG(H) =〈1〉. We study the groups, in which every non-normal subgroup is either contranormal or core-free. In particular, we obtain the structure of some monolithic and non-monolithic groups with this property

Finite groups with Frobenius condition for non-normal primary subgroups

A finite group P is said to be primary if for some prime p. We say a primary subgroup P of a finite group G satisfies the Frobenius condition in G if is a p-group provided P is p-group. In this article, we determine the structure of a finite group G in which every non-subnormal primary subgroup satisfies the Frobenius condition. In particular, we prove that if every non-normal primary subgroup of G satisfies the Frobenius condition, then is abelian and every maximal non-normal nilpotent subgroup U of G with is a Carter subgroup of G.

Isomorphism classes of transversals in finite nilpotent groups

In this article, we prove that if H is a non-normal subgroup of a finite nilpotent group G non-isomorphic to the dihedral group D 8 of order 8, then the number of isomorphism classes of normalized right transversals (NRTs) of H in G is greater than 16, where the isomorphism classes are formed with respect to the induced right-quasigroup structure (induced by the binary operation of G) on each NRT of H in G.

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 Non-Normal Subgroup Graph of Some Dihedral Groups

Let G be a finite groupand H is a subgroup of G. The subgroup graph of H in G is defined as a directed simple graph with vertex set G and two distinct elements x and y are adjacent if and only if xy∈H. In this paper, the work on subgroup graph is extended by defining a new graph called the non-normal subgroup graph. The subgroup graph is determined for some dihedral groupsof order 2n when the subgroup is non-normal. Keywords: subgroup graph; non-normal subgroup; dihedral group

Finite non-nilpotent groups with two conjugacy classes of non-normal non-cyclic subgroups

Finite non-nilpotent groups with two conjugacy classes of non-normal non-cyclic subgroups

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.