This preview shows page 1. Sign up to view the full content.
88
2. BASIC THEORY OF GROUPS
The statement (c) means, for example, that any group of order 3 is
isomorphic to
Z
3
. Statement (d) means that there are two distinct (noniso
morphic) groups of order 4, and any group of order 4 must be isomorphic
to one of them.
Proof.
The reader is guided through the proof of statements (a) through (e)
in the exercises. The idea is to try to write down the group multiplication
table, observing the constraint that each group element must appear exactly
once in each row and column.
Statements (a)–(e) give us the complete list, up to isomorphism, of
groups of order no more than 5, and we can see by going through the list
that all of them are abelian. Finally, we have already seen that
Z
6
and
S
3
are two nonisomorphic groups of order 6, and
S
3
is nonabelian.
n
The deﬁnition of isomorphism says that under the bijection, the multi
plication tables of the two groups match up, so the two groups differ only
by a renaming of elements. Since the multiplication tables match up, one
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
 Fall '08
 EVERAGE
 Algebra

Click to edit the document details