Attila Maroti
Rings as the unions of proper subrings.
Mon, 01/25/2010 - 16:01 — SomNambulAuthors: Attila Maroti, Andrea Lucchini
Subjects: Rings and Algebras
AbstractWe describe all possible ways how a ring can be expressed as the union of
three of its proper subrings. This is an analogue for rings of a 1926 theorem
of Scorza about groups. We then determine the minimal number of proper subrings
of the simple matrix ring $M_{n}(q)$ whose union is $M_{n}(q)$.Average dimension of fixed point spaces with applications.
Fri, 01/22/2010 - 16:01 — SomNambulAuthors: Robert M. Guralnick, Attila Maroti
Subjects: Group Theory
AbstractLet $G$ be a finite group, $F$ a field, and $V$ a finite dimensional
$FG$-module such that $G$ has no trivial composition factor on $V$. Then the
arithmetic average dimension of the fixed point spaces of elements of $G$ on
$V$ is at most $(1/p) \dim V$ where $p$ is the smallest prime divisor of the
order of $G$. This answers and generalizes a 1966 conjecture of Neumann which
also appeared in a paper of Neumann and Vaughan-Lee and also as a problem in
The Kourovka Notebook posted by Vaughan-Lee. Our result also generalizes a
recent theorem of Isaacs, Keller, Meierfrankenfeld, and Moret\'o.
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