SECTION 4.2 Basics of Group Theory 217
- LetAbe a nonempty set. For any non-empty setB⊆Afind the
inverse ofBwith respect to symmetic difference +. - 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?
- LetSbe a set and let F(S) be the set of all functionsf:S→S. Is
composition “◦” associative? Commutative? Is there an identity? - 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.