Transcription of A FRIENDLY INTRODUCTION TO GROUP THEORY
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A FRIENDLY INTRODUCTION TO GROUP THEORY
A FRIENDLY INTRODUCTION TO GROUP THEORYJAKE cares?You do, prefrosh. If you re a math major, then you probably want to pass Math you re a chemistry major, then you probably want to take that one chem class Iheard involves representation THEORY . If you re a physics major, then at some pointyou might want to know what the Standard Model is. And I ll bet at least a fewof you CS majors care at least a little bit about cryptography. Anyway, Wikipediathinks it s useful to know some basic GROUP THEORY , and I think I agree. It s also funand I promise it isn t very is a GROUP ?I m about to tell you what a GROUP is, so brace yourself for disappointment. It sbound to be a somewhat anticlimactic experience for both of us: I type out a bunchof unimpressive-looking properties, and a bunch of you sit there looking hope I can convince you, however, that it is the simplicity and ordinariness of thisdefinition that makes GROUP THEORY so deep and fundamentally 1:Agroup(G, )is a setGtogether with a binary operation :G G Gsatisfying the following three that is, for anyx,y,z G, we have(x y) z=x (y z).
A FRIENDLY INTRODUCTION TO GROUP THEORY 3 A good way to check your understanding of the above de nitions is to make sure you understand why the following equation is correct: jhgij= o(g): (1) De nition 5: A group Gis called abelian (or commutative) if gh = hg for all g;h2G. A group is called cyclic if it is generated by a single element, that is,
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