N-groups and fusion systems

Fusion systems for profinite groups

We introduce the notion of a pro-fusion system on a pro-p group, which generalizes the notion of a fusion system on a finite p-group. We also prove a version of Alperin’s Fusion Theorem for pro-fusion systems.

Some properties of n-capable and n-perfect groups

In this article we introduce the notion of n-capable groups. It is shown that every group G admits a uniquely determined subgroup (〖Z^n)〗^* (G) which is a characteristic subgroup and lies in the n-centre subgroup of the group G. This is the smallest subgroup of G whose factor group is n-capable. Moreover, some properties of n-central extension will be studied.

Saturated fusion systems over 2-groups

We develop methods for listing, for a given 2-group S, all nonconstrained centerfree saturated fusion systems over S. These are the saturated fusion systems which could, potentially, include minimal examples of exotic fusion systems: fusion systems not arising from any finite group. To test our methods, we carry out this program over four concrete examples: two of order 2 and two of order 2. Ou...

n-cyclicizer groups

A Characterization of Saturated Fusion Systems over Abelian 2-groups

Given a saturated fusion system F over a 2-group S, we prove that S is abelian provided any element of S is F-conjugate to an element of Z(S). As a consequence, if B is a 2-block of a finite group G such that all B-subsections are major, the defect groups of B are abelian. Our results were conjectured by Kühlshammer–Navarro–Sambale–Tiep [2] and generalize a Theorem of Camina–Herzog [3].