Extensions of finite nilpotent groups

A note on extensions of nilpotent groups

A further advance—though it was not presented as such—was contained in Theorem 1.6 of [1], which effectively gave an example showing that the answer to our question is not always affirmative. The main purpose of this paper is to generalize both Theorem 0.1 and the counterexample contained in Theorem 1.6 of [1]. Thus we prove (cf. Theorems 2.1 and 2.3, which reproduce part (b) and part (a) of Th...

Extensions of Nilpotent P.P. Rings

NILPOTENT p-LOCAL FINITE GROUPS

In this paper we provide characterizations of p-nilpotency for fusion systems and p-local finite groups that are inspired by known result for finite groups. In particular, we generalize criteria by Atiyah, Brunetti, Frobenius, Quillen, Stammbach and Tate.

Irredundant Generating Sets of Finite Nilpotent Groups

It is a standard fact of linear algebra that all bases of a vector space have the same cardinality, namely the dimension of the vector space over its base field. If we treat a vector space as an additive abelian group, then this is equivalent to saying that an irredundant generating set for a vector space must have cardinality equal to the dimension of the vector space. The same is not true for...

Nilpotent Groups of Finite Morley Rank

Now assume |G : H| is infinite and that H is definable in G (but H need no longer be infinite or G-normal). Let Z = Z◦(G), and note that (a) implies that Z is infinite. If |Z : H ∩Z| is infinite, then |HZ : H| is infinite as well. In this case we are done since H ≤ HZ ≤ NG(H). Otherwise, |Z : H ∩Z| is finite, so the connectedness of Z implies that Z ≤ H. We can now proceed by induction on the r...