Visual Algebra, Lecture 5.5: Actions by automorphisms скачать в хорошем качестве

Visual Algebra, Lecture 5.5: Actions by automorphisms

The automorphism group of any group naturally acts on the group itself, but also on the set of conjugacy classes. In this lecture, we’ll also look at related actions of the inner and outer automorphism groups. The full automorphism group also acts on the subgroups of G, and the orbits are at least as coarse as the conjugacy classes. The fixed points are called "characteristic subgroups," and this is stronger than normality. Examples of characteristic subgroups include the center, the commutator subgroup, and several new subgroups that we’ll see this lecture, such as the Frattini subgroup and the socle. We’ll interpret and explore these in terms of some of our favorite subgroup lattices. Course & book webpage (with complete lecture note slides, HW, exams, etc.): https://www.math.clemson.edu/~macaule... ------------------------------------------------------------------------------------------------------------------------------------------------------ 0:00 Introduction 0:54 Aut(Q₈) acting on Q₈ 5:38 Action graphs of Aut(Q₈) actions 9:35 Actions of Inn(Q₈) and Out(Q₈) 15:16 Characteristic subgroups 17:21 Subgroup automorph diagrams 22:28 The Frattini subgroup and the socle 26:50 Characteristic subgroups in Q₈⋊C₉ 30:44 Characteristic subgroups in SA₁₆ 35:25 Characteristic subgroups in AGL₁(ℤ₇)≅C₇⋊C₆