Related papers: Essential Dimension of Small Finite Groups
We discuss the notion of essential dimension of a finite group and explain its relation with birational algebraic geometry. We show how this leads to a (partial) classification of simple finite groups of essential dimension less than or…
We provide a simple method to compute upper bounds on the essential dimension of split reductive groups with finite or connected center by means of their generically free representations. Combining our upper bound with previously known…
We classify all finite groups of essential dimension 2 over an algebraically closed field of characteristic 0.
In this paper, we compute the essential $l$-dimension of the finite groups of classical Lie type for odd primes $l$ not equal to the defining prime, specifically the general linear groups, the symplectic groups, the orthogonal groups, and…
We classify quasi-simple finite groups of essential dimension 3.
We determine the finite groups whose real irreducible representations have different degrees.
We give a simple formula for the essential dimension of a finite pseudo-reflection group at a prime p and determine the absolute essential dimension for most irreducible pseudo-reflection groups. We also study the "poor man's essential…
Let K be an arbitrary field. We will determine explicitly all the nontrivial finite groups of essential dimension one over K.
We prove that if a finite group scheme $G$ over a field $k$ has essential dimension one, then it embeds in $PGL_{2/k}$. We use this to give an explicit classification of all infinitesimal group schemes of essential dimension one over any…
Given a finite smooth group scheme $G$ over a field of characteristic $p > 0$, we show that the essential dimension of $G$ at $p$ is $0$ when $p$ does not divide the order of $G$, and $1$ when it does.
We study the essential dimension of a finite group G over a field K. A generalization of the central extension theorem of Buhler and Reichstein (Compositio Math. 106 (1997) 159-179, Theorem 5.3) is obtained. We also get lower bounds of…
In this note, we study the finite groups with the number of cylic subgroups no greater than 6.
In this paper, we compute the essential $2$-dimension when the defining prime is odd of the general linear groups, the projective general linear groups, the special linear groups when $n$ is odd or $n = 2$, as well as the special linear…
A partial group with $n+1$ elements is, when regarded as a symmetric simplicial set, of dimension at most $n$. This dimension is $n$ if and only if the partial group is a group. As a consequence of the first statement, finite partial groups…
We describe structure of locally finite groups of finite centraliser dimension.
We determine the essential dimension of the spin group Spin(n) as an algebraic group over a field of characteristic 2, for n at least 15. In this range, the essential dimension is the same as in characteristic not 2. In particular, it is…
In this paper, we compute the essential $p$-dimension of the split finite quasi-simple groups of classical Lie type at the defining prime, specifically the quasi-simple groups arising from the general linear and special linear groups, the…
We discuss essential dimension of group schemes, with particular attention to infinitesimal group schemes. We prove that the essential dimension of a group scheme of finite type over a field k is at least equal to the difference between the…
We calculate the precise number of r-regular elements in the finite exceptional groups. As a corollary we find that the proportion of r-regular elements is at least 3577/18432 and for all \epsilon>0, there are infinitely finite simple…
We determine the groups of minimal order in which all groups of order n can embedded for 1 < n < 16. We further determine the order of a minimal group in which all groups or order n or less can be embedded, also for 1 < n < 16.