Advanced High-School Mathematics

(Tina Meador) #1

SECTION 4.2 Basics of Group Theory 217



  1. LetAbe a nonempty set. For any non-empty setB⊆Afind the
    inverse ofBwith respect to symmetic difference +.

  2. Let Matn(R) be then×nmatrices with real coefficients and define
    the binary operation∗by setting


A∗B=AB−BA,

whereA,B∈Matn(R). Is∗associative? Commutative? Is there
an identity?


  1. LetSbe a set and let F(S) be the set of all functionsf:S→S. Is
    composition “◦” associative? Commutative? Is there an identity?

  2. LetSbe a set and consider the set F(S,R) of all real-valued func-
    tions f : S → R. Define addition + and multiplication “·” on
    F(S,R) by the rules


(f+g)(s) = f(s) +g(s), (f·g)(s) = f(s)·g(s), s∈R.

Are these operations associative? Commutative? What about
identities? What about inverses?

4.2.4 The concept of a group


Let (G,∗) be a set together with a binary operation. We say that (G,∗)
is agroupif the following three properties hold:


∗is associative:that isg 1 ∗(g 2 ∗g 3 ) = (g 1 ∗g 2 )∗g 3 for allg 1 , g 2 , g 3 ∈G;
Ghas an identity: that is, there exists an elemente∈Gsuch that
e∗g=g∗e=e, for allg∈G;

Existence of inverses: that is, for every g ∈ G, there exists an
elementg′∈Gwith the property thatg′∗g=g∗g′=e.
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