Powerfully Nilpotent Groups of Class 2

Powerfully nilpotent groups

If a graph is cut along each edge and can be divided into many faces, the graph has been defined to be a map when each face is homomorphic to an open disk and a bipartite map called dessin. A dessin D is regular Aut (D) if action transfer on the edge set. In particular, given a finite group G, a regular dessin is uniquely regular dessin if there is only one isomorphism class of Aut (D) are isomorphic to G. In this paper, for a nilpotent automorphism group of odd prime power order and nilpotency class four, we employ group-theoretical methods to classify these uniquely regular dessins.

We study powerful 2-Engel groups. We show that every powerful 2-Engel group generated by three elements is nilpotent of class at most two. Surprisingly, the result does not hold when the number of generators is larger than three. In this article and its sequel, we classify powerful 2-Engel groups of class 3 that are minimal in the sense that every proper powerful section is nilpotent of class at most 2.

If ω ≡ 1 is a group law implying virtual nilpotence in every finitely generated metabelian group satisfying it, then it implies virtual nilpotence for the finitely generated groups of a large class of groups including all residually or locally soluble-or-finite groups. In fact the groups of satisfying such a law are all nilpotent-by-finite exponent where the nilpotency class and exponent in question are both bounded above in terms of the length of ω alone. This yields a dichotomy for words. Finally, if the law ω ≡ 1 satisfies a certain additional condition—obtaining in particular for any monoidal or Engel law—then the conclusion extends to the much larger class consisting of all ‘locally graded’ groups.

Some finitely presented nilpotent-by-abelian groups

Brian Hartley asked me whether a free (nilpotent of class 2 and exponent p2)-group of countable rank has a faithful linear representation of finite degree, p here being a prime of course. The answer is yes. The point is that this then yields via work of F. Leinen and M. J. Tomkinson, see [3,3.6] an image of a linear p-group, which is not even finitary linear. The question of which relatively free groups have faithful linear representations dates back at least to work of W. Magnus in the 1930's, see [4, pp. 33, 34 and the final comment on p. 40] for a discussion of this. Our construction, which works more generally, is a further contribution. We write ℜc for the variety of nilpotent groups of class at most c and (ℭ9 for the variety of groups of exponent dividing q.

Some results obtained by the calculations of the integral homology of free nilpotent Lie algebras Hi (L(x1, x2,. . . , xr)/γN+1) in the system for computational discrete algebra GAP are presented. The attention is focused on the occurrence of unexpected torsion in this homology, similar to one that arises for a 4-generated free nilpotent group of class 2. The main result is that even for the case of two generators, the torsion occurs in the fourth integral homology when the nilpotency class is 5. Moreover, only 7-torsion occurs, and no others. Namely, there is an isomorphism H4(L(x1, x2)/γ6) ≅ ℤ85 ⨁ ℤ /7ℤ. We also give an explicit mathematical proof of the fact that H4(L(x1, x2, x3, x4)/γ3) has nontrivial 3-torsion. Bibliography: 6 titles.

The study of homology groups ${\mathfrak {B}_n}(\Pi ,A),\mathfrak {B}$ a variety, $\Pi$ a group in $\mathfrak {B}$, and $A$ a suitable $\Pi$-module, is continued. A ’Tor’ is constructed which gives a better (but imperfect) approximation to these groups than a Tor previously considered. ${\mathfrak {B}_2}(\Pi ,Z)$ is calculated for various varieties $\mathfrak {B}$ and groups $\Pi$.

In this article, we study the growth of (fine) Selmer groups of elliptic curves in certain infinite Galois extensions where the Galois group G is a uniform, pro-p, p-adic Lie group. By comparing the growth of (fine) Selmer groups with that of class groups, we show that it is possible for the $$\mu $$ -invariant of the (fine) Selmer group to become arbitrarily large in a certain class of nilpotent, uniform, pro-p Lie extension. We also study the growth of fine Selmer groups in false Tate curve extensions.

P. Hall's classical equality for the number of conjugacy classes‎ ‎in $p$-groups yields $k(G) ge (3/2) log_2 |G|$ when $G$ is nilpotent‎. ‎Using only Hall's theorem‎, ‎this is the best one can do when $|G| = 2^n$‎. ‎Using a result of G.J‎. ‎Sherman‎, ‎we improve the constant $3/2$ to $5/3$‎, ‎which is best possible across all‎ ‎nilpotent groups and to $15/8$ when $G$ is nilpotent and $|G| ne 8,16$‎. ‎These results are then used to prove that $k(G) > log_3(|G|)$ when $G/N$ is nilpotent‎, ‎under ‎‎‎‎natural conditions on $N trianglelefteq G$‎. ‎Also‎, ‎when $G'$ is nilpotent of class $c$‎, ‎we prove that $k(G) ge (log |G|)^t$ when $|G|$ is large enough‎, ‎depending only on $(c,t)$‎.

The study of minimal irregular p-groups was initiated by Mann [9]. He defines such a group to be a finite irregular p-group in which every proper section is regular. He then deduces a large number of properties of such groups and shows, by construction, that there is no bound on their exponent. In this note methods involving varieties of groups are used to show that all minimal irregular p-groups can be ‘built up’ from cyclic groups, groups of exponent p and a relatively small class of metabelian minimal irregular p-groups. To phrase this more precisely, say that a group P is derivable from a family of groups {Qi} if P is a homomorphic image of a subdirect product of the Qi. Then we will prove as Theorem 2.2: a minimal irregular p-group of exponent pn and nilpotency class c is derivable from a family of three groups: i) a cyclic group of order pn, ii) a 2-generator monolithic group of exponent p and class c, iii) a minimal irregular p-group whi h is metabelian, has class p and has exponent p2.

We provide a description of the enveloping semigroup of the affine unipotent transformation T:X→X of the form T(x)=Ax+α, where A is a lower triangular unipotent matrix, α is a constant vector, and X is a finite-dimensional torus. In particular, we show that in this case the enveloping semigroup is a nilpotent group whose nilpotency class is at most the dimension of the underlying torus.

On the period of Fibonacci sequences in nilpotent groups

Finite groups and Lie rings with a metacyclic Frobenius group of automorphisms

JntertexTs, vol. 1» no. 2, Fall 1997 Inner Anxiety and Outward Exploration: The American Museum of Natural History and the Central Asiatic Expeditions Ronald Rainger T e x a s T e c h U n i v e r s i t y In the 1920s and 1930s, New York’s American Museum of Natural History initiated some of the most extensive and highly publicized explo¬ rations of the early twentieth century. Promoted largely by the institution’s president, Henry Fairfield Osborn, the museum sent scientists, explorers and wealthy business associates to “the ends of the earth” to record information and return with specimen collections for the vast repository on Central Park West.^ During the 1920s and 1930s, the museum sponsored Lincoln Ellsworth’s bid to be the first to fly over the North Pole (Amund¬ sen), Carl Akeley’s expeditions for elephants which he would later display in the museum’s Theodore Roosevelt Memorial (Akelcy; Haraway 26-58), and William ICing Gregory’s efforts to capture gorillas in the Congo (Gregory and Raven). Through the museum’s use of “incessant and nationwide publicity” (Wissler, 203), the public avidly followed the exploits of these and other explorers (Perkins). Among the museum’s most famous endeavors were the Central Asiatic Expeditions of the 1920s. Conceived and led by the museum scientist Roy Chapman Andrews, these were aseries of eight explorations between 1922 and 1930 to the Gobi Desert of Mongolia for the purpose of discovering the “missing link.” The expeditions received extensive coverage in peri¬ odicals and the popular press. In addition, Andrews was idealized as an intrepid explorer, adashing, fearless young gentleman pursuing adaring adventure in aremote, mysterious region of the world. Andrews reinforced that image through radio presentations, his popular book. On the Trail of Ancient Man^ and anumber of autobiographical accounts (Andrews, Ends of the Earth-, Lafollette 55, 57-58). Yet Andrews’s expeditions and personal ambitions embodied more than an interest in adventurous fieldwork and specimen collection. The Central Asiatic Expeditions as well as the museum’s other efforts were the product of aparticular culture. They reflected the ideas and objectives of an early twentieth-century New York elite of politicians, businessmen, scientists, and sportsmen who had close associations with the American Museum of Natural History. For those individuals’ expeditions to the Arctic, Central Africa or Asia constituted scientific analogues to America’s expanding 1 7 7 1 7 8 I N T E R T E X T S political and economic power. The museum’s expeditions were also ameans for preserving traditional ideals and values, for sustaining the power and prestige of aparticular class and ethnic group in asociety that was becoming increasingly urban, technological and pluralistic. As articulated by Osborn and actualized by Andrews, the museum’s outward ventures were an effort to quell inner anxieties among the museum’s and the nation’s elites. Those concerns and objectives reflected the museum’s social structure. By the early twentieth century, the American Museum was one of the largest institutions of its kind in the world. Although apublic museum, its trustees and administration were dominated by such powerful figures as the financier J.P. Morgan, the politician Joseph Choate, and railroad presidents E. H. Harriman and Cornelius Vanderbilt. Those individuals were not interested in science per se; rather the museum reflected their civic, philanthropic, and proprietary interests. As devout Protestants of Scottish and English heri¬ tage, these were men who had created the YMCA and served on the board of the Society for the Suppression of Vice. They helped establish the Metropolitan Museum of Art, the New York Public Librar>^ and other major civic cultural institutions (Satterlee; Mercer; Rainger 54-56). Originally the American Museum did not embody any specific policies or objectives, but given its leaders’ religious interests and concern for social welfare and social control, it was to serve as aplace where the urban masses could participate in an acceptable form of entertainment and could learn of nature’s bounty andmagnificence(Green). By the early twentieth century the museum’s leaders were emphasizing an additional theme: the presen'ation of the country’s flora, fauna, and physical environment. The commercial, industrial, and...