Subgroups defining automorphisms in locally nilpotent groups

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

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.

Do infinite nilpotent groups always have equipotent Abelian subgroups?

B. H. Neumann’s characterization of groups possessing a finite covering by proper subgroups and Baer’s characterization of groups with finite coverings by abelian subgroups are refined to results about finite coverings by normal subgroups. Various corollaries about the structure of groups having such finite coverings are derived. Using the method employed for the main theorem, a simplified proof of an earlier result of the third author concerning finite coverings by word subgroups is given.

B. H. Neumann’s characterization of groups possessing a finite covering by proper subgroups and Baer’s characterization of groups with finite coverings by abelian subgroups are refined to results about finite coverings by normal subgroups. Various corollaries about the structure of groups having such finite coverings are derived. Using the method employed for the main theorem, a simplified proof of an earlier result of the third author concerning finite coverings by word subgroups is given.

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].

On the normal indices of proper subgroups of finite groups

Let G be a finite group. A covering of G is a collection of proper subgroups of G whose union is equal to the entire G. If the number of proper subgroups in the covering is n, then the covering is called an n-covering. Considering that no group can be covered by two of its proper subgroups, n ≥ 3. An n-covering is called irredundant if no proper sub-collection of subgroups from the covering is able to cover G. If all members of an n-covering are maximal normal subgroups of G, then the covering is called a maximal n-covering. Let D be the intersection of all members of an n-covering. Then, the covering is said to have a core-free intersection if ∩g∈G gDg−1 = {1}. This paper characterizes nilpotent groups having a maximal irredundant 11-covering with a core-free intersection. It was found that a nilpotent group G has a maximal irredundant 11-covering with a core-free intersection if and only if it is isomorphic to (C2)10, (C3)5, (C3)6, (C5)3 or (C5)4.Let G be a finite group. A covering of G is a collection of proper subgroups of G whose union is equal to the entire G. If the number of proper subgroups in the covering is n, then the covering is called an n-covering. Considering that no group can be covered by two of its proper subgroups, n ≥ 3. An n-covering is called irredundant if no proper sub-collection of subgroups from the covering is able to cover G. If all members of an n-covering are maximal normal subgroups of G, then the covering is called a maximal n-covering. Let D be the intersection of all members of an n-covering. Then, the covering is said to have a core-free intersection if ∩g∈G gDg−1 = {1}. This paper characterizes nilpotent groups having a maximal irredundant 11-covering with a core-free intersection. It was found that a nilpotent group G has a maximal irredundant 11-covering with a core-free intersection if and only if it is isomorphic to (C2)10, (C3)5, (C3)6, (C5)3 or (C5)4.

The index [G:g] of the element g in the [finite] group G is the number of elements conjugate to -g in G. The significance of elements of prime power index is best recognized once one remembers Wielandt's Theorem that elements whose order and index are powers of the same prime p are contained in a normal subgroup of order a power of p and Burnside's Theorem asserting the absence of elements of prime power index, not 1, in simple groups. From Burnside's Theorem one deduces easily that a group without proper characteristic subgroups contains an element, not 1, whose index is a power of a prime if and only if this group is abelian. In this result it suffices to assume the absence of proper fully invariant subgroups, since we can prove [in ?2] the rather surprising result that a [finite] group does not possess proper fully invariant subgroups if and only if it does not possess proper characteristic subgroups. A deeper insight will be gained if we consider groups which contain many elements of prime power index. We show [in ?5 ] that the elements of order a power of p form a direct factor if, and only if, their indices are powers of p too; and nilpotency is naturally equivalent to the requirement that this property holds for every prime p. More difficult is the determination of groups with the property that every element of prime power order has also prime power index [?3]. It follows from Burnside's Theorem that such groups are soluble; and it is clear that a group has this property if it is the direct product of groups of relatively prime orders which are either p-groups or else have orders divisible by only two different primes and furthermore have abelian Sylow subgroups. But we are able to show conversely that every group with the property under consideration may be represented in the fashion indicated. In ?5 we study the so-called hypercenter. This characteristic subgroup has been defined in various ways: as the terminal member of the ascending central chain or as the smallest normal subgroup modulo which the center is 1. We may add here such further characterizations as the intersection of all the normalizers of all the Sylow subgroups or as the intersection of all the maximal nilpotent subgroups; and the connection with the index problem is obtained by showing that a normal subgroup is part of the hypercenter if, and only if, its elements of order a power of p have also index a powrer of p. Notation. All the groups under consideration will be finite. An element [group] is termed primary, if its order is a prime power;

Some finite solvable groups with no outer automorphisms

Some applications of graph theory to finite groups

The degree of an irreducible complex character afforded by a finite group is bounded above by the index of an abelian normal subgroup and by the square root of the index of the center. Whenever a finite group affords an irreducible character whose degree achieves these two upper bounds the group must be solvable. Let G be a finite group with an irreducible (complex) character ?. If Z is the center of G it is easy to prove that ~(1)2_ [G:Z] and if A is an abelian normal subgroup of G it is easy to show that t(1) ? [G:A A (see pp. 364-365 of [1]). Say the group G admits an irreducible character of large degree if t(1)2 = [G:A ]2 = [G:Z], that is whenever the two bounds noted above are achieved simultaneously. Such groups arise in the theory of projective representations and the Galois theory in general rings [2]. The purpose of this note is to give proof of the result stated in the title, thus verifying a special case of a conjecture in [2]. Througlhout all unexplained terminology is as in [1]. THEOREM. Let G be a group with center Z and abe/ian normal subgroup A so that there is an irreducible complex character v on G with t(1) 2 = [G: A ]j2= [G: Z]. Then G is solvable. PROOF. A theorem of P. Hall (Theorem 4.5, p. 233 of [31]) asserts that a group is solvable if and only if every p-sylow subgroup has a complement. This theorem will be applied to G/A to give the result. Since the degree of any irreducible character is bounded by the index of an abelian subgroup, A is a maximal abelian normal subgroup of G, so ZCA. If 7r, II, and P are sylow p-subgroups of Z, A, and G respectively then 7rCIICP. M\oreover wr is contained in the center of P, II is an abelian normal subgroup of P, and II is a normal subgroup of G. Arguing as in [2 ] we show p e = mX where X is irreducible on P and X(1) = [P:ll]. By Schur's lemma we have tJ z = r(1)q for some linear character $ of Z. Then (D, f G)(P Z, k) = (1) so by counting degrees D(1)v=OG. Let R be the subgroup of G generated by Z and P, and let X be an irreducible character of R contained in q1?. By Schur's Received by the editors December 29, 1969. AMS Subject Classifications. Primary 2080; Secondary 2027, 1670.

(2002). When Is a Group the Union of Proper Normal Subgroups? The American Mathematical Monthly: Vol. 109, No. 5, pp. 471-473.

In this study, the t-intuitionistic fuzzy normalizer and centralizer of t intuitionistic fuzzy subgroup are proposed. The t-intuitionistic fuzzy centralizer is normal subgroup of t-intuitionistic fuzzy normalizer and investigate various algebraic properties of this phenomena. We also introduce the concept of t-intuitionistic fuzzy Abelian and cyclic subgroups and prove that every t-intuitionistic fuzzy subgroup of Abelian (cyclic) group is t-intuitionistic fuzzy Abelian (cyclic) subgroup. We show that the image and pre-image of t-intuitionistic fuzzy Abelian (cyclic) subgroup are t-intuitionistic fuzzy Abelian (cyclic) subgroup under group homomorphism.

Suppose that G G is a nonabelian group with a unique proper normal subgroup of some given order. It is proved that G G is not contained as a normal subgroup within the Frattini subgroup of a finite p p -group.