Search Results

This is exercise 7.2.12 of Robinson's Course in the Theory of Groups, page 195 in the first edition. A transitive subgroup of prime degree is primitive, and primitive solvable groups have a regular n …

Here is a little survey of the "bottom" of finite groups. It includes an answer to (2) in the case of finite nilpotent groups, and a correction to an anonymous comment. The subgroup generated by the …

Lots of results in group cohomology only have topological proofs using the techniques of Bob Oliver and his (generalized) collaborators. For instance, many results along the lines of "controls fusion …

For a finite group $G$ is there a subgroup $H$ such that for every chief factor $K/L$ of $G$ one has: $G = K C_G(K/L)$ and $K \leq HL$ (so $K/L$ is inner and covered by $H$) $G \neq K C_G(K/L)$ and …

You might be considering a special case of the Schur-Zassenhaus theorem. If A is a normal Hall-subgroup of G, then A has a complement B, and all complements B are conjugate under the action of G. Th …

The numbers n such that every group of order n is cyclic, abelian, nilpotent, supersolvable, or solvable are known. Most are described in an easy to read survey: Pakianathan, Jonathan; Shankar, Kris …

The quotients of a finite group, G/N have minimal normal subgroups K/N. These are called chief factors. Chief factors are divided into two kinds, central chief factors and eccentric chief factors. …

Some of the other answers omit several details and I had already written this on another forum. At any rate, perhaps the example would be useful: More or less yes, though for larger groups it is not …

Short answer: a typical example is G=SL(2,5), H = Z(G) = Z/2Z. If G/H and H are coprime and satisfy the condition, the G = G/H × H is quite dull. I'll assume you find this interesting, and want …

This question is reasonably hard, but important. A very clear and explicit answer is given in: Flannery, D. L.; O'Brien, E. A. "Linear groups of small degree over finite fields." Internat. J. Algebr …

Sure, you just take direct products of metabelian groups with different character degrees: cd(G×H) = cd(G)×cd(H) and (G×H)′ = G′×H′. I suggest taking G(p) = AGL(1,p) = Hol(p) to be the normalizer of …

I don't know of a resource with such a list, though you could easily make one with programs such as GAP or Magma. Since it is reasonably easy and interesting to analyze this problem, I'll outline how …

I believe the holomorph of the elementary abelian group of order 2n is complete unless n=1 or 3. It is called AGL(n,2), the affine general linear group of dimension n over the field of size 2. In ot …

This is called a chief series or principal series. The quotients of the terms in the series are called chief factors, and the length of the series is called the chief length. Non-isomorphic groups c …

Feit published his paper in the proceedings of the first Jamaican conference, MR1484185. He defines M(n,K) to be the group of monomial matrices whose entries are roots of unity. M(n,Q) is the group …