15,853 research outputs found
The Large Davenport Constant I: Groups with a Cyclic, Index 2 Subgroup
Let be a finite group written multiplicatively. By a sequence over ,
we mean a finite sequence of terms from which is unordered, repetition of
terms allowed, and we say that it is a product-one sequence if its terms can be
ordered so that their product is the identity element of . The small
Davenport constant is the maximal integer such that
there is a sequence over of length which has no nontrivial,
product-one subsequence. The large Davenport constant is the
maximal length of a minimal product-one sequence---this is a product-one
sequence which cannot be factored into two nontrivial, product-one
subsequences. It is easily observed that , and
if is abelian, then equality holds. However, for non-abelian groups, these
constants can differ significantly. Now suppose has a cyclic, index 2
subgroup. Then an old result of Olson and White (dating back to 1977) implies
that if is non-cyclic, and
if is cyclic. In this paper, we determine the large Davenport constant of
such groups, showing that , where is the commutator subgroup of
Factorizations of finite groups by conjugate subgroups which are solvable or nilpotent
We consider factorizations of a finite group into conjugate subgroups,
for and ,
where is nilpotent or solvable. First we exploit the split -pair
structure of finite simple groups of Lie type to give a unified self-contained
proof that every such group is a product of four or three unipotent Sylow
subgroups. Then we derive an upper bound on the minimal length of a solvable
conjugate factorization of a general finite group. Finally, using conjugate
factorizations of a general finite solvable group by any of its Carter
subgroups, we obtain an upper bound on the minimal length of a nilpotent
conjugate factorization of a general finite group
A characterization of class groups via sets of lengths
Let be a Krull monoid with class group such that every class contains
a prime divisor. Then every nonunit can be written as a finite
product of irreducible elements. If , with
irreducibles , then is called the length of the
factorization and the set of all possible is called the set
of lengths of . It is well-known that the system depends only on the class group . In the present
paper we study the inverse question asking whether or not the system is characteristic for the class group. Consider a further Krull monoid
with class group such that every class contains a prime divisor and
suppose that . We show that, if one of the
groups and is finite and has rank at most two, then and are
isomorphic (apart from two well-known pairings).Comment: The current version is close to the one to appear in J. Korean Math.
Soc., yet it contains a detailed proof of Proposition 2.4. The content of
Chapter 4 of the first version had been split off and is presented in ' A
characterization of Krull monoids for which sets of lengths are (almost)
arithmetical progressions' by the same authors (see hal-01976941 and
arXiv:1901.03506
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