Abstract Algebra Part I: Group Theory

From last time: Let G be a set. A binary operation on G is a function m : G G G Some examples: Some non-examples Addition and multiplication Dot and scalar products on R n on Z, Q, R, C, Z >0. Subtraction on Z >0. Division on C. Multiplication on M n (R). Cross products on R. We get lazier and lazier as time goes on, writing binary operations with, or no symbol at all: Rule to follow: Be clear! ab = a b = m(a, b). Generically, we call binary operations products, i.e. Define a product on Z by a b = a.

From last time: A Group is a pair (G, ) consisting of a set G and a binary operation on G vocab: we say G is closed under. such that:. is associative.. There is an identity element e G. That is, e g = g = g e for any g G.. Every element of G has an inverse. That is, for any g G, there is an element g such that gg = e = g g. Examples: (R, ) The non-zero real numbers form a group under multiplication (GL n (C), ) The invertible matrices form a group under multiplication (Z/Z, +) The integers modulo form a group under addition

From last time: Theorem Let G be a group.. The identity element in G is unique. We denote this element by e.. For a given g G, g is unique.. (a ) = a.. For any x, y G, there is are unique elements g, g G so that xg = y and g x = y.. For any x, y G, (xy) = (y x ).

From last time: Theorem Let G be a group. MORE LAZINESS!. The identity element in G is unique. We denote this element by e.. For a given g G, g is unique.. (a ) = a.. For any x, y G, there is are unique elements g, g G so that xg = y and g x = y.. For any x, y G, (xy) = (y x ). Let G be a group means I have some fixed binary operation floating around in the background. Ambiguity: what do I call the set now?? The underlying set is the set of elements of G.

Powers Still, let G be a group... The associative property implies that for any x,... x n G, the value of x x... x n does not depend on on the expression is parenthesized. Define: Theorem For any x G, x n = x x... x and x n = (x n ). x m x n = x m+n for all integers m and n.

Order Still, let G be a group... What are some properties of groups? How are two groups similar or different?

Order Still, let G be a group... What are some properties of groups? How are two groups similar or different? Definition The order of G, denoted G, is the size of the underlying set. For any element x G, if x n = e for some n Z >0, we say the order of x is the smallest such n.

Catalog of groups. Z n, Q n, R n, C n under addition infinite. Q, R, C under multiplication infinite. Z/nZ under addition finite. (Z/nZ) = {a Z/nZ a is relatively prime to n} under multiplication. finite. M n (F ) under addition, where F = Q, R, C etc. infinite More to come:

Groups of symmetries

r r r r r s rs r s r s r s r s

Definition A presentation for a group G consists of a generating set S along with a set of relations R (equations using only the elements of S {e} and their inverses, establishing relationships) that are enough to completely determine the group structure of G. It is written generators relations.