Edith Adan-Bante
Let G be a finite solvable group and $χ\in \Irr(G)$ be a faithful character. We show that the derived length of G is bounded by a linear function of the number of distinct irreducible constituents of $χ\barχ$. We also discuss other properties of the decomposition of $χ\barχ$ into its irreducible constituents.
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Let $G$ be a finite solvable group and $χ, ψ\in \Irr(G)$ be complex characters of $G$. Let $α$ be an irreducible constituent of the product $χψ$. We show that the derived length of $\Ker(α)/\Ker(χψ)$ is bounded by a linear function on the number of distinct irreducible constituents of $χψ$
Howie and Duncan observed that a word in a free product with length at least two and which is not a proper power can be decomposed as a product of two cyclic subwords each of which is uniquely positioned. Using this property, they proved various important results about one-relator product of groups. In this paper, we show that similar results hold in a more general setting where we allow elements of order two.
Powers with minimal commutator length in free products of groups
该研究针对群自由积中不与自由因子元素共轭的元素的n次幂的极小换位子长度问题展开探究，并给出了详尽的解答。
Let $G$ be a supersolvable group and $A$ be a conjugacy class of $G$. Observe that for some integer $η(AA^{-1})>0$, $AA^{-1}=\{a b^{-1}\mid a,b\in A\}$ is the union of $η(AA^{-1})$ distinct conjugacy classes of $G$. Set ${\bf C}_G(A)=\{g\in G\mid a^g=a\text{for all} a\in A\}$. Then the derived length of $G/{\bf C}_G(A)$ is less or equal than $2η(A A^{-1})-1$.
The character degree product and the conjugacy length product for finite general linear groups
本文针对 K. Harada 关于有限群不可约特征标次数乘积整除共轭类长度乘积的猜想，在有限一般线性群与有限酉群上完成了验证。
We give an algorithm to compute stable commutator length in free products of cyclic groups which is polynomial time in the length of the input, the number of factors, and the orders of the finite factors. We also describe some experimental and theoretical applications of this algorithm.
Given groups $A$ and $B$, what is the minimal commutator length of the 2020th (for instance) power of an element $g\in A*B$ not conjugate to elements of the free factors? The exhaustive answer to this question is still unknown, but we can give an almost answer: this minimum is one of two numbers (simply depending on $A$ and $B$). Other similar problems are also considered.
We describe an effective version of the conjugacy problem and study it for wreath products and free solvable groups. The problem involves estimating the length of short conjugators between two elements of the group, a notion which leads to the definition of the conjugacy length function. We show that for free solvable groups the conjugacy length function is at most cubic. For wreath products the behaviour depends on the conjugacy length function of the two groups involved, as well as subgroup distortion within the quotient group.
The generalized Fitting height $h^*(G)$ of a finite group $G$ is the least number $h$ such that $\mathrm{F}_h^* (G) = G$, where $\mathrm{F}_{(0)}^* (G) = 1$, and $\mathrm{F}_{(i+1)}^*(G)$ is the inverse image of the generalized Fitting subgroup $\mathrm{F}^*(G/\mathrm{F}^*_{(i)} (G))$. Let $p$ be a prime, $1=G_0\leq G_1\leq\dots\leq G_{2h+1}=G$ be the shortest normal series in which for $i$ odd the factor $G_{i+1}/G_i$ is $p$-soluble (possibly trivial), and for $i$ even the factor $G_{i+1}/G_i$ is a (non-empty) direct product of nonabelian simple groups. Then $h=λ_p(G)$ is called the non-$p$-s…
While the finite-dimensional modules of the dihedral 2-groups over fields of characteristic 2 were classified over 30 years ago, very little is known about the tensor products of such modules. In this article, we compute the Loewy length of the tensor product of two modules of a dihedral two-group in characteristic 2. As an immediate consequence, we determine when such a tensor product has a projective direct summand.
We show that stable commutator length is rational on free products of free Abelian groups amalgamated over $\mathbb{Z}^k$, a class of groups containing the fundamental groups of all torus knot complements. We consider a geometric model for these groups and parameterize all surfaces with specified boundary mapping to this space. Using this work we provide a topological algorithm to compute stable commutator length in these groups. Further, we use the methods developed to show that in free products of cyclic groups the stable commutator length of a fixed varies quasirationally in the orders of t…
We describe algorithms and heuristics that allow us to express arbitrary elements of SLn (Z) and Sp2n (Z) as products of generators in particular "standard" generating sets. For elements obtained experimentally as random products, it produces product expressions whose lengths are competitive with the input lengths.
In this paper, we present a new method to construct solvable groups with derived length four and four character degrees. We then use this method to present a number of new families of groups with derived length four and four character degrees.
$p$-Groups with Derived Length Three and Three Character Degrees
本文构造了同时具有导长为三和三个特征标次数的p-群实例，为相关群论研究提供了具体例子。
In this article, we propose a geometric framework dedicated to the study of van Kampen diagrams of graph products of groups. As an application, we find information on the word and the conjugacy problems. The main new result of the paper deals with the computation of conjugacy length functions. More precisely, if $Γ$ is a finite graph and $\mathcal{G}= \{ G_u \mid u \in V(Γ) \}$ a collection of finitely generated groups indexed by the vertices of $Γ$, then $$\max\limits_{u \in V(Γ)} \mathrm{CLF}_{G_u}(n) \leq \mathrm{CLF}_{Γ\mathcal{G}}(n) \leq (D+1) \cdot n + \max\limits_{Δ\subset Γ\ \text{com…
In the present paper, for a pair $(G,N)$ of a group $G$ and its normal subgroup $N$, we consider the mixed commutator length $\mathrm{cl}_{G,N}$ on the mixed commutator subgroup $[G,N]$. We focus on the setting of wreath products: $ (G,N)=(\mathbb{Z}\wr Γ, \bigoplus_Γ\mathbb{Z})$. Then we determine mixed commutator lengths in terms of the general rank in the sense of Malcev. As a byproduct, when an abelian group $Γ$ is not locally cyclic, the ordinary commutator length $\mathrm{cl}_G$ does not coincide with $\mathrm{cl}_{G,N}$ on $[G,N]$ for the above pair. On the other hand, we prove that if …
Algorithmic recognition of quasipositive 4-braids of algebraic length three
本文基于Garside理论提出多项式时间算法，识别代数长度为3的拟正4-辫子，并将两因子共轭幂判定推广到任意股数。
Let $F$ be a free group. We present for arbitrary $g\in\mathbb{N}$ a LogSpace (and thus polynomial time) algorithm that determines whether a given $w\in F$ is a product of at most $g$ commutators; and more generally an algorithm that determines, given $w\in F$, the minimal $g$ such that $w$ may be written as a product of $g$ commutators (and returns $\infty$ if no such $g$ exists). The algorithm also returns words $x_1,y_1,\dots,x_g,y_g$ such that $w=[x_1,y_1]\cdots[x_g,y_g]$. The algorithms we present are also efficient in practice. Using them, we produce the first example of a word in the fr…
Every finite group $G$ has a normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. We define the nonsoluble length $λ(G)$ as the minimum number of nonsoluble factors in a series of this kind. Upper bounds for $λ(G)$ appear in the study of various problems on finite, residually finite, and profinite groups. We prove that $λ(G)$ is bounded in terms of the maximum $2$-length of soluble subgroups of $G$, and that $λ(G)$ is bounded by the maximum Fitting height of soluble subgroups. For an odd prime $p$, the non-$p$-soluble length $λ_p(G)$ is intr…
On minimal product-one sequences of maximal length over the non-abelian group of order $pq$
本文刻画了阶为$pq$的非阿贝尔群$C_q \rtimes C_p$上所有长度为$2q$的极小乘积一序列，并将结果应用于研究该群上乘积一序列幺半群的$k$次弹性及相关集合的并集。
Finiteness properties of Subgroups of Houghton Groups of full Hirsch length
研究Houghton群中具有完全Hirsch长度的子群的有限性性质，证明当n≥3时，所有存在满同态到Z^{n-1}的大子群属于F_{n-1}型而非FP_n型，并引入广义置换圈积来描述这些子群。
The stable torsion length in a group is the stable word length with respect to the set of all torsion elements. We show that the stable torsion length vanishes in crystallographic groups. We then give a linear programming algorithm to compute a lower bound for stable torsion length in free products of groups. Moreover, we obtain an algorithm that exactly computes stable torsion length in free products of finite groups. The nature of the algorithm shows that stable torsion length is rational in this case. As applications, we give the first exact computations of stable torsion length for nontriv…
Sign-twisted generating functions of the odd length for Weyl groups of type $D$
本文研究$D$型Weyl群抛物商的奇数长度符号扭曲生成函数，建立显式闭乘积公式，并验证Brenti–Carnevale的三个猜想、解决Stembridge猜想。
We show that if $C_1$ and $C_2$ are directed cycles (of length at least two), then the Cartesian product $C_1 \Box C_2$ has two arc-disjoint hamiltonian paths. (This answers a question asked by J. A. Gallian in 1985.) The same conclusion also holds for the Cartesian product of any four or more directed cycles (of length at least two), but some cases remain open for the Cartesian product of three directed cycles. We also discuss the existence of arc-disjoint hamiltonian paths in $2$-generated Cayley digraphs on (finite or infinite) abelian groups.
A subset of a group is characteristic if it is invariant under every automorphism of the group. We study word length in fundamental groups of closed hyperbolic surfaces with respect to characteristic generating sets consisting of a finite union of orbits of the automorphism group, and show that the translation length of any element with nonzero crossing number is positive, and bounded below by a constant depending only (and explicitly) on a bound on the crossing numbers of generating elements. This answers a question of B. Farb.
Each element of the commutator subgroup of a group can be represented as a product of commutators. The minimal number of factors in such a product is called the commutator length of the element. The commutator length of a group is defined as the supremum of commutator lengths of elements of its commutator subgroup. We show that for certain closed symplectic manifolds, including complex projective spaces and Grassmannians, the universal cover of the group of Hamiltonian symplectomorphisms of the manifold has infinite commutator length. In particular, we present explicit examples of elements in …
Word Length and Diameter in Permutation Groups
本文研究置换群中二元直径与长度问题的计算复杂性，证明直径问题为PSPACE完全，在2-步幂零群中分别为Π₂^P完全和NP完全。
The reflection length of an element of a Coxeter group is the minimal number of conjugates of the standard generators whose product is equal to that element. In this paper we prove the conjecture of McCammond and Petersen that reflection length is unbounded in any non-affine Coxeter group. Among the tools used, the construction of word-hyperbolic quotients of all minimal non-affine Coxeter groups might be of independent interest.
Bounded cohomology and scl of verbal wreath products
本文研究了verbal缠绕积Γ≀_W A的有界上同调和稳定交换长度，证明其在某些情况下稳定交换长度为零，有界上同调在正度数上消失，包括标准限制缠绕积及来自n-可解、n-幂等和k-Burnside( k=2,3,4,6)的verbal缠绕积。
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