Abstract Algebra-The Basic Graduate Year (1).pdf

page 1 of Chapter 1
CHAPTER 1 GROUP FUNDAMENTALS
Groups and Subgroups
Definition A group is a nonempty set G on which there is defined a binary operation
(a, b) → ab satisfying the following properties:
Closure:Ifa and b belong to G, then ab is also in G;
Associativity: a(bc)=(ab)c for all a, b, c ∈ G;
Identity: There is an element 1 in G such that a1=1a = a for all a in G;
Inverse:Ifa is in G there is an element a−1 in G such that aa−1 = a−1a =1.
A group G is abelian if the binary operation mutative, ., ab = ba for all a, b in
G. In this case the binary operation is often written additively ((a, b) → a + b)), with the
identity written as 0 rather than 1.
There are some very familiar examples of abelian groups under addition, namely the
integers Z, the rationals Q, the real numbers R, plex numbers C, and the integers
Zm modulo m. Nonabelian groups will begin to appear in the next section.
The associative law generalizes to products of any finite number of elements, for ex-
ample, (ab)(cde)=a(bcd)e. A formal proof can be given by induction: if two people A
and B form a1 ···an in different ways, the last multiplication performed by A might look
like (a1 ···ai)(ai+1 ···an), and the last multiplication by B might be (a1 ···aj)(aj+1 ···an).
But if (without loss of generality) i<j, then (induction hypothesis)
(a1 ···aj)=(a1 ···ai)(ai+1 ···aj)
and
(ai+1 ···an)=(ai+1 ···aj)(aj+1 ···an).
By the n = 3 case, ., the associative law as stated in the definition of a group, the products
computed by A and B are the same.
The identity is unique (1 =11 = 1), as is the inverse of any given element (if b and
b are inverses of a then b =1b =(ba)b = b(ab)=b1=b). Exactly the same argument
shows that if b is a right inverse, and b a left inverse, of a, then b = b.
Definitions ments A subgroup H of a group G is a nonempty subset
of G that forms a group under the binary operation of G. Equivalently, H is a nonempty
subset of G such