The classification for -generator -groups of class has been completed by Morse, Magidin and Ahmad. Using this classification, we give formulas for the number and size of conjugacy classes of these groups.
UNIVERSITI SAINS MALAYSIA
azhana79 at yahoo dot com
A subgroup of a group is a solitary subgroup of if does not contain another isomorphic copy of . A normal subgroup of a group is a normal solitary subgroup of if does not contain another normal isomorphic copy of . A group is (normal) solitary solvable if it has a sub-(normal) solitary series with abelian.
WESTERN CAROLINA UNIVERSITY
ratanasov at email dot wcu dot edu
We define a -group, , to be normally serially monomial if there exists a single normal series,
such that for , and for every Irr , there exists with and Irr , such that is linear and . We investigate the character theoretic properties of such groups and the relation of the character degrees (and their multiplicities) to the group theoretic structure. Specifically, we show for ,
TEXAS STATE UNIVERSITY - SAN MARCOS
SAN MARCOS, TX
timwbonner at gmail dot com
A subgroup of a group satisfies the Frattini argument in provided for each subgroup normal in , . Examples of subgroups which satisfy the Frattini argument are injectors for a Fitting set. In a current project with J. Evan and S. Reifferscheid, we have obtained that if , and satisfies the Frattini argument in , then the diagonal sections (projection mod intersection with coordinate) are nilpotent but need not be abelian.
ben at math dot binghamton dot edu
In this talk I will present some observations about the derived length of a Coxeter group. In particular, I will give necessary and sufficient conditions for a Coxeter group to be "almost perfect". This is a report on current joint work with A. Piggott.
pbrooksb at bucknell dot edu
The Alperin weight conjecture - which proposes that the number of Alperin weights of a finite group is equal to the number of -regular conjugacy classes of , where is a prime, is known to be true for . However, in the original proof of Alperin and Fong, no explicit bijection is given between the two sets. Since the -regular conjugacy classes of are indexed by the -regular partitions of , then it would be nice to find an explicit bijection from the -regular partitions of to the Alperin weights of . While we are not yet able to do this, we can construct an explicit bijection from a related set of partitions to the Alperin weights of .
UNIVERSITY OF AKRON
cossey at uakron dot edu
In his 1943 TAMS paper, "Projective Planes", Marshall Hall Jr., introduced the idea of a ternary ring. In this talk, is a ternary ring if has distinct elements 1 and 0 with mapping into such that:
T2. for all in .
Following M. Hall Jr., using , we may introduce binary operations , on so that and are groupoids with identity 0,1 respectively. We call a ternary division ring if and are quasigroups with identity 0,1 respectively. We briefly discuss the double pointed categories TrnR and TrnDR of ternary rings and ternary division rings respectively. Finally, we focus our attention on a fixed ternary division ring and its groups.
DEPARTMENT OF MATHEMATICS, WESTERN MICHIGAN UNIVERSITY
clifton dot e.ealy at wmich dot edu
Given a finite group , how many squares are possible in a set of mutually orthogonal latin squares based on ? This question has been answered for elementary abelian groups, groups of small order, and groups with nontrivial, cyclic Sylow -subgroups. We will describe lower bounds for the number of squares possible in sets of mutually orthogonal latin squares based on nonabelian groups.
WRIGHT STATE UNIVERSITY
anthony dot evans at wright dot edu
A -group is a group all of whose subnormal subgroups are normal. It is possible to define a -group, one in which all -subnormal subgroups are normal, where is a formation of solvable groups locally defined by a formation function with appropriate properties. If is the formation of nilpotent groups, the -groups are just the -groups, whereas if is the formation of solvable groups, the -groups are just the Dedekind groups. This talk will describe possibilities for the property of being a -group when is between these extremes, investigating how to tell whether distinct formations yield distinct such properties.
FRANKLIN & MARSHALL COLLEGE
afeldman at fandm dot edu
A subgroup of a group is a solitary subgroup of if does not contain another isomorphic copy of . A normal subgroup of a group a normal solitary subgroup of if does not contain another normal isomorphic copy of . A group is (normal) solitary solvable if is has a sub-(normal) solitary series with abelian.
In this talk we will look at finite (normal) solitary solvable groups.
WESTERN CAROLINA UNIVERSITY
tsfoguel at wcu dot edu
We'll consider the connection between conjugation in groups and semigroups with inverse semigroups. This is work in progress.
fer at math dot binghamton dot edu
Let and be groups acting on each other and acting on themselves by conjugation, where and for and . We say the mutual actions are compatible if
Compatible actions play a role in the nonabelian tensor product defined as follows.
Little is known about compatible actions. This is due in part to the fact that little is known about the automorphism groups in general. But even if we know the automorphism groups, like in the case of cyclic groups, our knowledge consists of fragments.
The topic of this talk is to shed some light on the mystery of compatible actions. We will give a brief overview on what is known so far, provide some new results in case of cyclic groups, and discuss various approaches on how to unravel this mystery further.
menger at math dot binghamton dot edu
In this talk, we will go over some results about groups in which all subgroups are permutable or of finite rank. We will show the solubility of these type of groups in certain classes. Also we will obtain a bound for the cases when these groups are soluble.
UNIVERSITY OF ALABAMA
yzkaratas at crimson dot ua dot edu
We present a lower bound for the number of conjugacy classes of a finite group in terms of the largest prime divisor of the group order. We also present examples for which this bound is best possible. It is conjectured that these examples are the only ones meeting this bound, and we discuss recent progress on this conjecture (joint work with Hethelyi, Horvath, Maroti).
TEXAS STATE UNIVERSITY
SAN MARCOS, TX
tk04 at txstate dot edu
Let G be an infinite group and let Sub(G) := . (Sub(G), ) forms a lattice. We can then look at the set of all closure operators on Sub(G) ( (Sub(G)), which also forms a lattice. In this talk we will take a look at possible properties, including an algebraic property, of the lattice (Sub(G).
kilpack at math dot binghamton dot edu
Cayley-Dickson loop (L,*) is a loop of basis units of an algebra constructed by Cayley-Dickson doubling process (the first few examples of such algebras are complex numbers, quaternions, octonions, sedenions). We will discuss properties of Cayley-Dickson loops, identities they satisfy and the structure of their automorphism groups.
UNIVERSITY OF DENVER
ykirshte at du dot edu
In group theory, examining a group and studying its subgroups and their respective properties is interesting. In this talk, I will state a very exciting result which characterizes a containment of subgroups in a direct product. I will also give an application of this result; specifically the construction of the lattice of .
dlewis5 at binghamton dot edu
Let be a solvable group and let be a -Brauer character of where is an odd prime. We say is a lift of if is the restriction of to the -regular elements of .
We show that the generalized vertices for are all conjugate and if is a generalized vertex for , then is a vertex for and is linear. With this result in hand, we show that if has an abelian vertex , then has at most lifts. Finally, we discuss what is needed to prove this for any vertex . (This is joint work with James P. Cossey.)
KENT STATE UNIVERSITY
lewis at math dot kent dot edu
Bob Oliver conjectures that if is an odd prime and is a finite -group, then the Thompson subgroup is contained in a certain characteristic subgroup , which is now known as the Oliver subgroup. This conjecture would imply the existence and uniqueness of centric linking systems for fusion systems over odd primes.
Oliver's conjecture has a module-theoretic reformulation due to Green, Hethelyi, and Lillienthal. The main question arising out of this reformulation is the following: for an odd prime, a finite -group and an , does the presence of certain ``large" quadratic subgroups in force the existence of quadratic elements in the center of ? I will present recent work on this question which settles Oliver's conjecture in a couple of special cases, including for of nilpotence class at most roughly the (base 2) logarithm of .
THE OHIO STATE UNIVERSITY
jlynd at math dot ohio-state dot edu
If and are groups, the -nilpotent product of and is defined to be , where is the free product of and , and is the st term of the lower central series of . Golovin proved in the 1950s that every element of the -nilpotent product can be written uniquely as , where , , and , the latter called the cartesian of the nilpotent product. In 1960, MacHenry proved that the cartesian of the -nilpotent product is isomoprhic to via the map .
Using a construction introduced by Rocco to compute the nonabelian tensor square of a group, we show that if , then the cartesian of the -nilpotent product of with itself has the nonabelian tensor square of as a quotient. We explore this connection, with the hope of getting insight into both the tensor square and the cartesian of a -nilpotent product.
UNIVERSITY OF LOUISIANA AT LAFAYETTE
magidin at member dot ams dot org
If the nonabelian tensor square of a group is abelian then the derived subgroup of is abelian. This follows from the fact that an epimorphism from to always exists. Hence those groups whose nonabelian tensor squares abelian are metabelian. However, the converse is not true. Let is free nilpotent of class 3. Then is metabelian but is nilpotent of exactly class 2 (Blyth, Moravec, Morse, 2008). The purpose of this paper is to precisely define the subclass of metabelian groups whose nonabelian tensor squares are abelian.
SULTAN IDRIS EDUCATION UNIVERSITY (UPSI)
TANJONG MALIM, PERAK, MALAYSIA
rohaidah at fst dot upsi dot edu dot my
The Bogomolov multiplier is a group theoretical invariant isomorphic to the unramified Brauer group of a given quotient space, and represents an obstruction to the problem of stable rationality. We describe a homological version of the Bogomolov multiplier, find a five term exact sequence corresponding to this invariant, and describe the role of the Bogomolov multiplier in the theory of central extensions. We define the Bogomolov multiplier within K-theory and show that proving its triviality is equivalent to solving a problem posed by Bass. An algorithm for computing the Bogomolov multiplier is presented.
UNIVERSITY OF LJUBLJANA
primoz dot moravec at fmf dot uni-lj dot si
In this talk we consider the capability of -groups of nilpotency class 2 of exponent and -groups in which and is elementary abelian of rank 2.
UNIVERSITY OF EVANSVILLE
rfmorse at evansville dot edu
This talk is a part of my work about conjugacy of
subgroups of a finite group
containing a nilpotent subgroup of class
of maximal order.
We study the operation of groups
(the set of
nilpotent subgroups of class
order) on the set of components of
Then we get the important result:
Theorem. Let be a finite group, , a component of with and .
UNIVERSITY OF TÜBINGEN
av dot neumann at mymail dot ch
We present conditions on the structure and degree of a finite irreducible complex linear group that guarantee its solvability. In particular, we show that if such a group is -solvable but not -closed for some prime number , then the group is solvable whenever and is also smaller than certain bounds which are on the order of .
newtonb at beloit dot edu
According to Bernhard Neumann, every group with a noncyclic finite homomorphic image is the union of finitely many proper subgroups. The minimal number of subgroups needed to cover a group is called the covering number of . Tomkinson showed that for a solvable group the covering number is of the form prime power plus one and he suggested the investigation of the covering number for families of finite simple groups. So far, a few results are known, among them some for small alternating groups, several types of linear groups, and the Suzuki groups. For sporadic simple groups the covering numbers are known for the Matthieu groups and , as well as Ly and O'N. Furthermore, estimates have been given for J1 and McL. We have started to determine the covering number for other sporadic simple groups such as the Matthieu group .
FLORIDA ATLANTIC UNIVERSITY
BOCA RATON, FL
nikolova20032003 at yahoo dot com
Let be a saturated fusion system on a finite -group , where is a prime. Following an approach developed by Stellmacher for finite groups, I will define a certain positive characteristic functor . As a counterpart for the prime to Glauberman's -theorem, Stellmacher proved that any nontrivial -group has a nontrivial characteristic subgroup with the following property. For any finite -free group , with a Sylow -subgroup of and with self-centralizing, the subgroup is normal in . I will show how to generalize Stellmacher's result to fusion systems. For odd primes, the functor is used to generalize Thompson's normal complement theorem to fusion systems. This is work done in collaboration with Radu Stancu.
THE OHIO STATE UNIVERSITY
onofrei at math dot ohio-state dot edu
In their article, ``On the derived subgroup of the free nilpotent groups of finite rank'', Russell Blyth, Primoz Moravec, and Robert Morse apply the work of S. Moran's ``A subgroup theorem for free nilpotent groups'' to discuss the structure of the derived subgroups of the free nilpotent groups of finite rank. Specifically, they construct an isomorphism for such a derived subgroup in terms of a direct product of a nonabelian group and a free abelian group. Having accomplished this feat, they then apply this result to computing the nonabelian tensor squares of the free nilpotent groups of finite rank.
In this talk, we discuss expansions of this research to investigate the structure of all of the other members of the lower central series and of the derived series of a free nilpotent group of finite rank.
SAINT LOUIS UNIVERSITY
SAINT LOUIS, MO
mark dot pedigo at gmail dot com
In this talk, we will calculate the number of subgroups in a direct product of finite cyclic groups by applying the fundamental theorem of finite abelian groups and a well-known structure theorem due to Goursat. We will also suggest ways in which the results can be generalized to a direct product of arbitrary finite groups.
petrillo at alfred dot edu
Analyzing the properties possessed by a nonassociative Moufang loop of order 81, we show that such a loop could be either of exponent three or nine. The general product rule of such a loop is governed by four parameters for the first case, and three for the second. Then by studying all possible values of these parameters, we find that there exist only five non-isomorphic cases: three of exponent three and two of exponent nine.
SCHOOL OF MATHEMATICAL SCIENCES, UNIVERSITI SAINS MALAYSIA
andy at cs dot usm dot my
Let be a prime and let be a Sylow -subgroup of the symmetric group of degree . The group contains an "easy to see" normal subgroup that is elementary abelian of rank . For the subgroups of having a unique minimal normal subgroup, our goal is to calculate the number of faithful irreducible ordinary characters of of each degree. Part of our motivation is that under certain circumstances the number of such characters of of degree is related to the order of the automorphism group . We have achieved our goal for all subgroups satisfying all of the following conditions: (1) splits over its normal subgroup , (2) the factor group has exponent , and (3) the order of is where is divisible by .
UNIVERSITY OF AKRON
riedl at uakron dot edu
A generalized Baumslag-Solitar group is the fundamental group of a graph of groups with infinite cyclic vertex and edge groups. A method is described for calculating the Schur mutiplier of an arbitrary group of this type.
UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN
dsrobins at illinois dot edu
The holomorph of a cyclic group of odd prime power order is a good example of a group in which subnormal subgroups permute with all Sylow system normalizers, but may not permute with the Sylow subgroups themselves. This example is also indicative of the full classification, at least amongst those groups whose nilpotent residual has nilpotency class at most two. This is joint work with Jim Beidleman and Hermann Heineken.
UNIVERSITY OF KENTUCKY
jack at ms dot uky dot edu
This talk will present limits on the possible sets of irreducible character degrees of a normally monomial 5-group of maximal class.
mikes at mscs dot mu dot edu
We introduce a generalization of the non-abelian tensor product. Let and be groups which act on each other and which act on themselves. The actions of and are said to be compatible, if for all . The box-tensor product is defined provided and act on each other compatibly. In such a case is the group generated by the symbols with relations and , for all and . Note that if the groups act on themselves by conjugation, then the box tensor product is the nonabelian tensor product. In this talk we give a general construction for the box-tensor product . We describe the box-tensor product as a subgroup of a quotient of the free product . Such a construction was given by N R Rocco as well as Ellis and Leonard indepen dently for the nonabelian tensor product. We will discuss some examples of box-tensor product.
vthomas at math dot binghamton dot edu
Let be a finite group. Traditionally, the representation theory of considers a commutative ring , and then studies modules over on which acts where the actions of and on obey some compatibility properties. When is instead a commutative -ring, one defines similarly -modules over , where the compatibility now takes into account the action of on . The traditional notion of module over a ring is then simply a -module over the -ring , where we take the action of on to be trivial. The author has defined the notion of the Brauer-Clifford group of certain -algebras over fields. The Brauer-Clifford group is useful for the study of Clifford theory of finite groups, and in particular, it has been used to prove a strengthened version of the McKay Conjecture for all finite -solvable groups. We see how, by incorporating the study of -modules over commutative -rings, we may give a natural definition of the Brauer-Clifford group of -rings. This simpler definition extends the definition of the Brauer-Clifford group, and provides a more flexible basis for applications to the Clifford theory of finite groups.
UNIVERSITY OF FLORIDA
turull at ufl dot edu
This is a brief and incomplete overview of open and recently solved problems in loop theory (loops are ``nonassociative groups'') in which group theory plays a crucial role. I will speak about automorphic loops, loops with commuting inner mappings, and Moufang loops.
UNIVERSITY OF DENVER
petr at math dot du dot edu
A covering of a group is a collection of subgroups of so that for all , there exists an so that . Whenever consists of only abelian subgroups, there is a natural way to associate a ring with the cover , namely by defining for each . Here the are just functions and the operations are function addition and composition of functions. We are interested in how properties of this ring can influence structural properties of the group .
In this paper we consider the situation in which is a p-group and is a covering by groups which are elementary abelian of order . We can associate a graph with each such cover. This graph can then be used to determine properties of the ring, in particular, it can be used to decide when the ring is simple.
As a consequence, we are able to show that unless a p-group of exponent has an element whose centralizer has order , there will exist a covering of this group for which the ring is not simple.
This theorem leads to a general discussion of p-groups which have an element of order whose centralizer has order . It turns out that all such groups must be of maximal class.
SOUTHEASTERN LOUISIANA UNIVERSITY
gary dot walls at selu dot edu
Any Cayley table is a Latin square, two-thirds of being a Sudoku-like table. Cayley-Sudoku tables are Cayley tables arranged in such a way as to satisfy the additional Sudoku requirement, namely, that the Cayley table is divided into rectangular blocks with each group element appearing exactly once in each block. A recreational math project with undergraduates on constructing Cayley-Sudoku tables led to this potentially interesting question about transversals. Given a subgroup of a (finite) group , under what circumstances is it possible to partition into sets where for every each is a left transversal of ? After some remarks on Cayley-Sudoku tables for motivation, we present what we know about the transversal question from a group theoretic and then a combinatorial perspective, with the hope that some listener will know more.
WESTERN OREGON UNIVERSITY
wardm at wou dot edu
We'll discuss the consequences of knowing that the base of a finite permutational wreath product is not characteristic. Much has been discovered in this topic in the case of standard wreath products and wreath products where acts transitively, but we'll discuss finite wreath products where the action of is only faithful. The talk will develop an understanding of centralizers in a wreath product from a new viewpoint and examine the number of conjugates with which a given element may commute. This will ultimately lead to the conclusion that if the base of a finite permutational wreath product is not characteristic then must equal where is an odd order abelian group on which , an element of order 2, acts by inversion.
wilcox at math dot binghamton dot edu
We prove a structure theorem for the isometry group of an Hermitian map , where and are vector spaces over a finite field of odd order. We also present a Las Vegas polynomial-time algorithm to find generators for this isometry group, and to determine its structure. The algorithm can be adapted to construct the intersection of the members in a set of classical subgroups of , yielding the first polynomial-time solution of this old problem. Our approach develops new computational tools for algebras with involution, which in turn have applications to other algorithmic problems of interest. An implementation of our algorithm in the Magma system demonstrates its practicability.
THE OHIO STATE UNIVERSITY
wilson at math dot ohio-state dot edu
We settle a conjecture by Walter Carlip. Suppose is a finite solvable group, is a faithful -module over a field of characteristic and assume . Let be a nilpotent subgroup of and assume that involves no wreath product for or a Mersenne prime, then has at least one regular orbit on .
TEXAS STATE UNIVERSITY AT SAN MARCOS
SAN MARCOS, TX
yang at txstate dot edu