Search Results

The converse is not true in general. Beyl, Felgner and Schmid presented a condition in which the capability of a direct product of finitely many of groups implies the capability of each of the factor …

Presentations for the Siegel modular groups $Sp(2g,\mathbb{Z})$ are closely related to presentations of the surface mapping class group.In general, the braid group $B_{2g+2}$ acts on the free group $F …

A good reference is the book "Finite group Theory" of M. Aschbacher. Chapter 11 is devoted to the generalized Fitting subgroup $F^*(G)$ of $G$, and is quite detailed. The second one, $\tilde{F}(G)$ ha …

Classical groups are really a classical topic and have been studied a lot. It is therefore not easy to find some interesting topics which are still accessible. Certainly there are some aspects of unit …

Frobenius groups have trivial center. This is helpful for non-examples, e.g., to see that $p$-groups, or the quaternion group is not Frobenius (and all nilpotent groups). Further examples are: non-abe …

All free groups of finite or infinite countable rank are subgroups of the free non-abelian group $F_2$, which is linear. However, a free group of infinite uncountable rank will not be a subgroup of $F …

The Wielandt-Kegel theorem states that all finite dinilpotent groups G=AB are solvable. This results extends to many infinite groups, e.g., to finitely-generated linear groups. In general however, I t …

Every discrete group of Euclidean isometries acts properly discontinuously and cocompactly on some subspace of $\mathbb{R}^n$ (a result of Bieberbach). If we are in the crystallographic case, then the …

I found explicit results for $nse(G)$ for many finite simple groups $G$. Indeed, there is a large literature on the characterisation of finite simple groups $G$ by their order $\mid G\mid$ and the set …

Yes, in general $L$ need not be a prime knot, if $K$ is a prime knot, given a knot epimorphism $f\colon \pi K \to \pi L$, see section $4$ of D.S. Silver and W. Whitten, Knot group epimorphisms II (whi …

The mentioned result of Cohn has been further extended. Let us write $σ(G) = n$ whenever $G$ is the union of $n$ proper subgroups, but is not the union of any smaller number of proper sub- groups. Thu …

For finite groups, solvability can be detected from Engel-like identities. This was not really the question, but it is very interesting in this context, I think. The proof is surprisingly complicated …

This is an exercise in many topology books. Here is a reference with a complete proof: Look up Example 9.15 in the book "Knots" by G. Burde and H. Zieschang. The Jacobian of the presentation $G(T_{p,q …

There are some results for special cases. Burnside has proved in $1912$ that, if $G$ is a non-metabelian $p$-group, then the centre of the derived group of $G$ cannot be cyclic. In particular, a non- …

For a finite group $G$, denote the maximum of the orders of its Abelian subgroups by $a(G)$. Then we have, for $G=F_4(q)$ and $q$ even, $$ q^{11}\le a(G)\le q^{17}, $$ and for $G=F_4(q)$, $q$ odd, $$ …