Nilpotent Maximal Subgroup Research Articles

Nilpotent covers of symmetric and alternating groups

We prove that the symmetric group $S_n$ has a unique minimal cover $\mathcal{M}$ by maximal nilpotent subgroups, and we obtain an explicit and easily computed formula for the size of $\mathcal{M}$. In addition, we prove that the size of $\mathcal{M}$ is equal to the size of a maximal non-nilpotent subset of $S_n$. This cover $\mathcal{M}$ has attractive properties; for instance, it is a normal cover, and the number of conjugacy classes of subgroups in the cover is equal to the number of partitions of $n$ into distinct positive integers. We show that these results contrast with those for the alternating group $A_n$. In particular, we prove that, for all but finitely many values of $n$, no minimal cover of $A_n$ by maximal nilpotent subgroups is a normal cover and the size of a minimal cover of $A_n$ by maximal nilpotent subgroups is strictly greater than the size of a maximal non-nilpotent subset of $A_n$.

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.

Unipotent representations of Lie incidence geometries

Given a flag \(A\) of a spherical building \(\Delta \) of rank \(n\ge 3\), let \(\Gamma := \Delta _{J,A}\) be the residue of \(A\) in the \(J\)-shadow geometry \(\Delta _J\) of \(\Delta \), where \(J := \tau (A)\) is the type of \(A\). Let \(A^*\) be a flag opposite to \(A\) and \(G_{A^*}\) the stabilizer of \(A^*\) in the group \(G := \mathrm {Aut}(\Delta )\) of all type-preserving automorphisms of \(\Delta \). It is well known that \(G_{A^*}\) admits a maximal normal nilpotent subgroup \(U_{A^*}\), called the unipotent radical of \(G_{A^*}\) and acting regularly on the set of flags opposite to \(A^*\). As we will show in this paper, a representation \(\varepsilon _\Delta ^J\) of \(\Gamma \) in \(U_{A^*}\) can be given, every element of \(\Gamma \) being mapped onto a suitable subgroup of \(U_{A^*}\), containment between elements of \(\Gamma \) corresponding to containment between the corresponding subgroups. We call such a representation a unipotent representation. We develop some theory for unipotent representations and we examine a number of interesting cases, where a projective embedding of \(\Gamma \) can be obtained as a quotient of \(\varepsilon _\Delta ^J\) by factorizing over the derived subgroup of \(U_{A^*}\), while \(\varepsilon ^J_\Delta \) itself is dominant, namely it is not a proper image of any other representation of \(\Gamma \) in any group.

Nilpotent and polycyclic-by-finite maximal subgroups of skew linear groups

Let D be an infinite division ring, n a natural number and N a subnormal subgroup of GLn(D) such that n=1 or the center of D contains at least five elements. This paper contains two main results. In the first one we prove that each nilpotent maximal subgroup of N is abelian; this generalizes the result in Ebrahimian (2004) [3] (which asserts that each maximal subgroup of GLn(D) is abelian) and a result in Ramezan-Nassab and Kiani (2013) [12]. In the second one we show that a maximal subgroup of GLn(D) cannot be polycyclic-by-finite.

NILPOTENT AND LOCALLY FINITE MAXIMAL SUBGROUPS OF SKEW LINEAR GROUPS

Let D be a division ring and N be a subnormal subgroup of D*. In this paper we prove that if M is a nilpotent maximal subgroup of N, then M′ is abelian. If, furthermore every element of M is algebraic over Z(D) and M′ ⊈ F* or M/Z(M) or M′ is finitely generated, then M is abelian. The second main result of this paper concerns the subgroups of matrix groups; assume D is a noncommutative division ring, n is a natural number, N is a subnormal subgroup of GLn(D), and M is a maximal subgroup of N. We show that if M is locally finite over Z(D)*, then M is either absolutely irreducible or abelian.

Some sufficient conditions on the number of non-abelian subgroups of a finite group to be solvable

Thompson’s theorem indicates that a finite group with a nilpotent maximal subgroup of odd order is solvable. As an important application of Thompson’s theorem, a finite group is solvable if it has an abelian maximal subgroup. In this paper, we give some sufficient conditions on the number of non-abelian subgroups of a finite group to be solvable.

On Locally Nilpotent Subgroups of GL 1(D)

In this article we study locally nilpotent subgroups of D*: = GL 1(D), where D is a division ring. It is proved that every locally nilpotent subnormal subgroup of D* is central. If D is algebraic over its centre then every locally solvable subnormal subgroup of D* is central. Also, in this case, it is shown that every locally nilpotent maximal subgroup of D* can occur as the multiplicative group of some maximal subfield of D.

Nilpotent maximal subgroups of GLn( D)

In [S. Akbari, J. Algebra 217 (1999) 422–433] it has been conjectured that if D is a noncommutative division ring, then D ∗ contains no nilpotent maximal subgroup. In connection with this conjecture we show that if GL n ( D) contains a nilpotent maximal subgroup, say M, then M is abelian, provided D is infinite. This extends one of the main results appeared in [S. Akbari, J. Algebra 259 (2003) 201–225, Theorem 4].

Maximal subgroups of GLn(D)

In this paper we study the structure of locally solvable, solvable, locally nilpotent, and nilpotent maximal subgroups of skew linear groups. In [S. Akbari et al., J. Algebra 217 (1999) 422–433] it has been conjectured that if D is a division ring and M a nilpotent maximal subgroup of D ∗ , then D is commutative. In connection with this conjecture we show that if F [ M ]⧹ F contains an algebraic element over F , then M is an abelian group. Also we show that C ∗ ∪ C ∗ j is a solvable maximal subgroup of real quaternions and so give a counterexample to Conjecture 3 of [S. Akbari et al., J. Algebra 217 (1999) 422–433], which states that if D is a division ring and M a solvable maximal subgroup of D ∗ , then D is commutative. Also we completely determine the structure of division rings with a non-abelian algebraic locally solvable maximal subgroup, which gives a full solution to both cases given in Theorem 8 of [S. Akbari et al., J. Algebra 217 (1999) 422–433]. Ultimately, we extend our results to the general skew linear groups.

On finitary linear groups with a locally nilpotent maximal subgroup

We prove that a group G of finitary permutations, containing a locally nilpotent maximal subgroup M is locally solvable if M is not a 2-group. We also prove that the same is true if G is a periodic, non-modular, finitary linear group.

Product of nilpotent subgroups

We will say that a subgroup X of G satisfies property C in G if \({\rm C}_{G}(X\cap X^{{g}})\leqq X\cap X^{{g}}\) for all \({g}\in G\). We obtain that if X is a nilpotent subgroup satisfying property C in G, then XF(G) is nilpotent. As consequence it follows that if \(N\triangleleft G\) is nilpotent and X is a nilpotent subgroup of G then \(C_G(N\cap X)\leqq X\) implies that NX is nilpotent.¶We investigate the relationship between the maximal nilpotent subgroups satisfying property C and the nilpotent injectors in a finite group.

More on b-injectors of sporadic groups+

A B-Injector in an arbitrary finite group G is defined as a maximal nilpo-tent subgroup of G, containing a subgroup A of G of maximal order satisfying class(A) ≥ 2. Among other results the B-Injectors of the sporadic groups J1,J2,J4, M24 are determined.

A partition of characters associated to nilpotent subgroups

IfG is a finite solvable group andH is a maximal nilpotent subgroup ofG containingF(G), we show that there is a canonical basisP(G|H) of the space of class functions onG vanishing off anyG-conjugate ofH which consists of characters. ViaP(G|H) it is possible to partition the irreducible characters ofG into “blocks”. These behave like Brauerp-blocks and a Fong theory for them can be developed.

Locally finite groups with a nilpotent or locally nilpotent maximal subgroup

Locally finite groups with a nilpotent or locally nilpotent maximal subgroup

Nilpotent injectors in finite groups

In [3] A. Bialostocki has introduced a special class of maximal nilpotent subgroups in an arbitrary finite group. ¶In this paper we want to investigate the relationship between this special class of maximal nilpotent subgroups and the more classical $\cal N $ -injectors in an arbitrary finite group. ¶To do this we prove the following theorem which is of fundamental importance in this investigation: ¶Let U be a nilpotent subgroup of G containing every U-invariant nilpotent subgroup of G. Then U is an $\cal N $ -injector of G.

On local-solvability of groups containing a locally nilpotent maximal subgroup

i, Introduction. In this paper we study groups with a locally nilpotent maximal subgroup. Our purpose is to find which conditions on the group G, or on the maximal locally nilpotent subgroup M, ensure the local solvability of G. tn the theory of finite groups there is a well-known result of Thompson, which says that if G is a finite group with a maximal subgroup M which is nilpotent and of odd order, then G is solvable (see [9]). The restriction on the order of M was due to the fact that, in order to obtain the above result, Thompson uses his famous J Theorem to prove the p-nilpotency of G/M~ (i,e. the existence of a normal p-complement) for the primes p involved in M / M G (M o is the core of M in G). Thus p must be # 2. Deskins and Janko proved that G is solvable even when the order of M is even, provided the Sylow 2-subgroup of M has class at most 2 (see [2] and [6]), The general concern with the Sylow 2-subgroup of M is due to the existence of a counterexample: namely PSL (2, I7), which is simple and contains a maximal nilpotent subgroup isomorphic to D16, hence a 2-subgroup of nilpotency class 3. In the case that G is an infinite periodic linear group contained in GL (n, K), Wehrfritz proved that if M is a maximal subgroup of G which is locally nilpotent and if the Sytow p-subgroup of M is finite or regular for p = char K, p 4= 2, while the Sylow 2-subgroup is nilpotent Of class at most 2, then G is solvable ([I0], 12,8). Wehrfritz says that the hypothesis on the Sylow p-subgroup of M, for p char K, is probably redundant: we prove this in Section 3, Some of the above results were extended to locally finite groups by Bruno and Schuur (see [1]),

Nilpotent Injectors in General Linear Groups

In a recent paper, A. Bialostocki (Israel J. Math.41 (1982), 261-273) has defined a nilpotent injector in an arbitrary finite group G to be a maximal nilpotent subgroup of G, containing a subgroup H of G of maximal order satisfying class (H) ≤2. In the present paper, the author determines the nilpotent injectors of GL(n, q) and shows that they form a unique conjugacy class of subgroups of GL(n, q). It is also proved that if n ≠ 2 or n = 2 and q ≠ 9 is not a Fermat prime >3, then the nilpotent injectors of GL(n, q) are the nilpotent subgroups of maximal order.

On locally finite groups with a locally nilpotent maximal subgroup

On locally finite groups with a locally nilpotent maximal subgroup

Maximal supersoluble subgroups of symmetric groups

Some thirty-five years ago, Suprunenko proved that every symmetric group S~ possesses a unique conjugacy class of maximal (with respect to inclusion) nilpotent transitive subgroups, and also the intransitive maximal nilpotent subgroups of Sn can be classified (see [3]). The purpose of this paper is the classification of all maximal supersoluble subgroups in symmetric groups. In contrast to the nilpotent case, there is usually more than one conjugacy class of maximal supersoluble transitive subgroups in S~; even more, there is no bound (independent of n) on the number of those conjugacy classes. The proof of the main result in this paper splits into two parts. In Section 4 we present a certain general construction of maximal supersoluble transitive subgroups of a given symmetric group Sn and in Section 5 we show that there are no others. The construction depends on certain ,~admissible, factorizations of n and on the choice of a certain regular abelian group. Different admissible factorization of n yield different conjugacy classes of maximal supersoluble transitive subgroups of S~; whether there are any non-trivial admissible factorizations of n (the trivial ones are n = n. 1 if n is odd and n = (n/2). 2 if n is even) depends on the primes involved in n. Exactly when there are odd prime divisors p and q of n such that q divides p 1 or when 4 divides n and n is not a power of 2, there exist a non-trivial admissible factori-

Lower bounds for the number of conjugacy classes in finite solvable groups

We prove first that if G is a finite solvable group of derived length d ≥ 2, then k(G) > |G|1/(2d−1), where k(G) is the number of conjugacy classes in G. Next, a growth assumption on the sequence [G(i): G(i+1)] 1 d−1 , where G(i) is theith derived group, leads to a |G|1/(2d−1) lower bound for k(G), from which we derive a |G|c/log 2log2|G| lower bound, independent of d(G). Finally, “almost logarithmic” lower bounds are found for solvable groups with a nilpotent maximal subgroup, and for all Frobenius groups, solvable or not.