Introduction to Sporadic Groups

Introduction to Sporadic Groups

This is an introduction to finite simple groups, in particular sporadic groups, intended for physicists. After a short review of group theory, we enumerate the 1 + 1 + 16 = 18 families of finite simple groups, as an introduction to the sporadic groups. These are described next, in three levels of increasing complexity, plus the six isolated “pariah” groups. The (old) five Mathieu groups make up...

Sporadic isogenies to orthogonal groups

SO(p, q) = {g ∈ SLp+q(R) : g>Qg = Q} (where Q = ( 1p 0 0 −1q ) )  SU(2) −→ SO(3) SL2(R) −→ SO(2, 1) SU(2)× SU(2) −→ SO(4) SL2(C) −→ SO(3, 1) SL2(R)× SL2(R) −→ SO(2, 2) Sp∗(2, 0) −→ SO(5) Sp∗(1, 1) −→ SO(4, 1) Sp2(R) −→ SO(3, 2) SU(4) −→ SO(6) SL2(H) −→ SO(5, 1) SU(2, 2) −→ SO(4, 2) SL4(R) −→ SO(3, 3) Thus, these are small examples...

Introduction to arithmetic groups

Arithmetic groups are fundamental groups of locally symmetric spaces. We will see how they are constructed, and discuss some of their important properties. For example, although the Q-rank of an arithmetic group is usually defined in purely algebraic terms, we will see that it provides important information about the geometry and topology of the corresponding locally symmetric space. Algebraic ...

Introduction to Lie groups

Introduction to Transformation Groups

These are the notes from a course that was an introduction to the theory of finite (or compact) transformation groups, including basic definitions and facts, standard examples, algebraic restrictions, fundamental constructions, all with an emphasis on group actions on manifolds in lowdimensional geometric topology. There is a focus on problems and results that have interesting versions in low d...