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Summary: Dec. 2007 Sultan Almuhammadi ICS 555 LNGT
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Lecture Notes on Group Theory
§ 1. Introduction to Groups
[1] Definition. A group (G, ·) is a nonempty set G together with a binary operation · on
G such that the following conditions hold:
(i) Closure: For all a,b G the element a · b is a uniquely defined element of G
(ii) Associativity: For all a,b,c G, we have
a · (b · c) = (a · b) · c
(iii) Identity: There exists an identity element e G such that for all a G
e · a = a and a · e = a
(iv) Inverses: For each a G there exists an inverse element a-1
G such that
a · a-1
= e and a-1
· a = e
[2] Notations.
1. Juxtaposition: we usually write "ab" for the product (a · b)
2. Power (Superscript): an
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