Divisibility Graph for Symmetric and Alternating Groups

non-divisibility for abelian groups

Throughout all groups are abelian. We say a group G is n-divisible if nG = G. If G has no non-zero n-divisible subgroups for all n>1 then we say that G is absolutely non-divisible. In the study of class C consisting   all absolutely non-divisible groups such as G, we come across the sub groups T_p(G) = the sum of all p-divisible subgroups and rad_p(G) = the intersection of all p^nG. The proper...

Some finite groups with divisibility graph containing no triangles

Let $G$ be a finite group. The graph $D(G)$ is a divisibility graph of $G$. Its vertex set is the non-central conjugacy class sizes of $G$ and there is an edge between vertices $a$ and $b$ if and only if $a|b$ or $b|a$. In this paper, we investigate the structure of the divisibility graph $D(G)$ for a non-solvable group with $sigma^{ast}(G)=2$, a finite simple group $G$ that satisfies the one-p...

Short presentations for alternating and symmetric groups

We derive new families of presentations (by generators and relations) for the alternating and symmetric groups of finite degree n. These include presentations of length that are linear in log n, and 2-generator presentations with a bounded number of relations independent of n.

Regular polytopes with symmetric and alternating groups

On Alternating and Symmetric Groups as Galois Groups

Fix an integer n = 3. We show that the alternating group An appears as Galois group over any Hilbertian field of characteristic different from 2. In characteristic 2, we prove the same when n is odd. We show that any quadratic extension of Hilbertian fields of characteristic different from 2 can be embedded in an Sn–extension (i.e. a Galois extension with the symmetric group Sn as Galois group)...