We study the strong, weak, and special amalgamation bases in the varieties of nilpotent groups of class two and exponent n, where n is odd. The main result is a characterization of the special amalgamation bases for these varieties. We also characterize the weak and strong bases. For special amalgamation bases, we show that there are groups which are special bases in varieties of finite exponent but not in the variety of all nil-2 groups, whereas for weak and strong bases we show this is not the case. We also show that in these varieties, as well as the variety of all nil-2 groups, a group has an absolute closure (in the sense of Isbell) if and only if it is already absolutely closed, i.e. if and only if it is a special amalgamation base.
We give necessary and sufficient conditions for weak and strong embeddability of amalgams in each subvariety of the category of all nilpotent groups of class at most two; this generalizes B. Maier's result for the latter class. We also discuss dominions (in the sense of Isbell), and characterize the weak, strong, and special amalgamation bases for each subvariety, contrasting the resulting classes with one another.
Let $G$ be an odd order nilpotent group with class 2 and $e$ denotes the exponent of its commutator subgroup. Let $e=p_1^{r_1}p_2^{r_2}... p_s^{r_s}$, where $p_i$'s are odd primes and $r_i$'s are non-negative integers. Then there are at least $r_1+r_2+... +r_s$ non-isomorphic nilpotent groups with class two and the order of each of the group is equal to the order of $G$.
Suppose that $A$ is a finite nilpotent group of odd order acting good in the sense of \cite{EGJ} on the group $G$ of odd order. Under some additional assumptions we prove that the Fitting height of $G$ is bounded above by the sum of the numbers of primes dividing $|A|$ and $|C_G(A)|$ counted with multiplicities.
First, I construct an isomorphism between the categories of (topological) groups of nilpotency class 2 with 2-divisible center and (topological) Lie rings of nilpotency class 2 with 2-divisible center. That isomorphism allows us to construct adjoint and coadjoint representations as usual. For a finite group G of nilpotency class 2 of odd order, I construct a basis in its group algebra C[G], parameterized by elements of g* so that the elements of coadjoint orbits form bases of simple two-side ideals of C[G]. That construction gives us a one-to-one correspondence between G-orbits in g* and class…
For any odd prime $p$, we give an example of a locally finite $p$-group $G$ containing a left 3-Engel element $x$ where $\langle x \rangle^G$ is not nilpotent.
We introduce a special class of powerful $p$-groups that we call powerfully nilpotent groups that are finite $p$-groups that possess a central series of a special kind. To these we can attach the notion of a powerful nilpotence class that leads naturally to a classification in terms of an `ancestry tree' and powerful coclass. We show that there are finitely many powerfully nilpotent $p$-groups of each given powerful coclass and develop some general theory for this class of groups. We also determine the growth of powerfully nilpotent groups of exponent $p^{2}$ and order $p^{n}$ where $p$ is odd…
We prove that if $p$ is an odd prime, $G$ is a solvable group, and the average value of the irreducible characters of $G$ whose degrees are not divisible by $p$ is strictly less than $2(p+1)/(p+3)$, then $G$ is $p$-nilpotent. We show that there are examples that are not $p$-nilpotent where this bound is met for every prime $p$. We then prove a number of variations of this result.
Let $G$ be a group and write $\mbox{Perm}(G)$ for its symmetric group. Define $\mbox{Hol}(G)$ to be the holomorph of $G$, regarded as a subgroup of $\mbox{Perm}(G)$, and let $\mbox{NHol}(G)$ denote its normalizer. The quotient $T(G) = \mbox{NHol}(G)/\mbox{Hol}(G)$ has been computed for various families of groups $G$, and in most of the known cases, it turns out to be elementary $2$-abelian, except for two groups of order $16$ and some groups of odd prime power order and nilpotency class two. In this paper, we shall show that $T(G)$ is elementary $2$-abelian for all finite groups $G$ of squaref…
A finite group $G$ is called {\it $p^i$-central of height $k$} if every element of order $p^i$ of $G$ is contained in the $k^{th}$-term $ζ_k(G)$ of the ascending central series of $G$. If $p$ is odd such a group has to be $p$-nilpotent (Thm. A). Finite $p$-central $p$-groups of height $p-2$ can be seen as the dual analogue of finite potent $p$-groups, i.e., for such a finite $p$-group $P$ the group $P/Ω_1(P)$ is also $p$-central of height $p-2$ (Thm. B). In such a group $P$ the index of $P^p$ is less or equal than the order of the subgroup $Ω_1(P)$ (Thm. C). If the Sylow $p$-subgroup $P$ of a …
In this paper we prove a group theoretic analogue of the well known local nilpotence theorem for sandwich Lie algebras due to Kostrikin and Zel'manov. We introduce the notion of a strong left 3-Engel element of a group G and show that these are always in the locally nilpotent radical of G. This generalises a previous result of Jabara and Traustason that showed that a left 3-Engel element a of a group G is in the locally nilpotent radical of G whenever a is of odd order.
A longstanding problem attributed to I. Schur says that for a finite group $G$, the exponent of the second homology group $H_2(G, \mathbb{Z})$ divides the exponent of $G$. In this paper, we prove this conjecture for finite nilpotent groups of odd exponent and of nilpotency class 5, $p$-central metabelian $p$ groups, and groups considered by L. E . Wilson in \cite{LEW}. Moreover, we improve several bounds given by various authors.
Let $G$ be a finite group admitting a coprime automorphism $α$ of order $e$. Denote by $I_G(α)$ the set of commutators $g^{-1}g^α$, where $g\in G$, and by $[G,α]$ the subgroup generated by $I_G(α)$. We study the impact of $I_G(α)$ on the structure of $[G,α]$. Suppose that each subgroup generated by a subset of $I_G(α)$ can be generated by at most $r$ elements. We show that the rank of $[G,α]$ is $(e,r)$-bounded. Along the way, we establish several results of independent interest. In particular, we prove that if every element of $I_G(α)$ has odd order, then $[G,α]$ has odd order too. Further, i…
In this note we show that if $p$ is an odd prime and $G$ is a powerful $p$-group with $N\leq G^{p}$ and $N$ normal in $G$, then $N$ is powerfully nilpotent. An analogous result is proved for $p=2$ when $N\leq G^{4}$.
Let $G$ be a finite group of odd order admitting an involutory automorphism $ϕ$. We obtain two results bounding the exponent of $[G,ϕ]$. Denote by $G_{-ϕ}$ the set $\{[g,ϕ]\,\vert\, g\in G\}$ and by $G_ϕ$ the centralizer of $ϕ$, that is, the subgroup of fixed points of $ϕ$. The obtained results are as follows.1. Assume that the subgroup $\langle x,y\rangle$ has derived length at most $d$ and $x^e=1$ for every $x,y\in G_{-ϕ}$. Suppose that $G_ϕ$ is nilpotent of class $c$. Then the exponent of $[G,ϕ]$ is $(c,d,e)$-bounded.2. Assume that $G_ϕ$ has rank $r$ and $x^e=1$ for each $x\in G_{-ϕ}$. Then…
A linear odd Poisson bracket realized solely in terms of Grassmann variables is suggested. It is revealed that with the bracket, corresponding to a semi-simple Lie group, both a Grassmann-odd Casimir function and invariant (with respect to this group) nilpotent differential operators of the first, second and third orders are naturally related and enter into a finite-dimensional Lie superalgebra. A connection of the quantities, forming this Lie superalgebra, with the BRST charge, $Δ$-operator and ghost number operator is indicated.
It is a longstanding conjecture that for a finite group $G$, the exponent of the second homology group $H_2(G, \mathbb{Z})$ divides the exponent of $G$. In this paper, we prove this conjecture for $p$-groups of class at most $p$, finite nilpotent groups of odd exponent and of nilpotency class 5, $p$-central metabelian $p$-groups, and groups considered by L. E . Wilson in \cite{LEW}. Moreover, we improve several bounds given by various authors. We achieve most of our results using an induction argument.
We study the groups $G$ with the curious property that there exists an element $k\in G$ and a function $f\colon G\to G$ such that $f(xk)=xf(x)$ holds for all $x\in G$. This property arose from the study of near-rings and input-output automata on groups. We call a group with this property a $J$-group. Finite $J$-groups must have odd order, and hence are solvable. We prove that every finite nilpotent group of odd order is a $J$-group if its nilpotency class $c$ satisfies $c\le6$. If $G$ is a finite $p$-group, with $p>2$ and $p^2>2c-1$, then we prove that $G$ is $J$-group. Finally, if $p>2$ and $…
We show that for a wide class of groups of finite Morley rank the presence of a split $BN$-pair of Tits rank $1$ forces the group to be of the form $\operatorname{PSL}_2$ and the $BN$-pair to be standard. Our approach is via the theory of Moufang sets. Specifically, we investigate infinite and so-called hereditarily proper Moufang sets of finite Morley rank in the case where the little projective group has no infinite elementary abelian $2$-subgroups and show that all such Moufang sets are standard (and thus associated to $\operatorname{PSL}_2(F)$ for $F$ an algebraically closed field of chara…
Let E be the extraspecial p-group of order p^3 and exponent p where p is an odd prime. We determine the mod p cohomology of summands in the stable splitting of p-completed classifying space BE modulo nilpotence.
The main theorem in this article shows that a group of odd order which admits the alternating group of degree 5 with an element of order 5 acting fixed point freely is nilpotent of class at most two. For all odd primes r, other than 5, we give a class two r-group which admits the alternating group of degree 5 in such a way. This theorem corrects an earlier result which asserts that such class two groups do not exist. The result allows us to state a theorem giving precise information about groups in which the centralizer of every element of order 5 has centralizer a 5-group.
Let S be a p-group for an odd prime p. Bob Oliver conjectures that a certain characteristic subgroup X(S) always contains the Thompson subgroup J(S). We obtain a reformulation of the conjecture as a statement about modular representations of p-groups. Using this we verify Oliver's conjecture for groups where S/X(S) has nilpotence class at most two.
There is a longstanding conjecture, due to Gregory Cherlin and Boris Zilber, that all simple groups of finite Morley rank are simple algebraic groups. One of the major theorems in the area is Borovik's trichotomy theorem. The "trichotomy" here is a case division of the minimal counterexamples within odd type, i.e. groups with a divisibble connected component of the Sylow 2-subgroup. We introduce a charateristic zero notion of unipotence which can be used to obtain a connected nilpotent signalizer functor from any sufficiently non-trivial solvable signalizer functor. This result plugs seamlessl…
