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The Nuts and Bolts of Proofs, 4th Edition
book

The Nuts and Bolts of Proofs, 4th Edition

by Antonella Cupillari
January 2011
Beginner content levelBeginner
296 pages
11h 43m
English
Academic Press
Content preview from The Nuts and Bolts of Proofs, 4th Edition
Isomorphism and Subgroups
One of the goals mentioned earlier was to investigate when groups of the same size have the same structure
andwhentheydonot.Lets revisit it. It wa s eas y t o co nclude that a ll g roups of size one have the same
structure, because the only element present must be the identity element.
All groups of size two have the same structure, and they a re abelian. Let s explicitly compare
ðP
2
, Þ= S
2
, ð
2
, +
2
Þ,andð
3
, ×
3
Þ: The set P
2
has two elements, t =
12
12

and f =
12
21

.The
set
2
has two elements, [0]
2
and [1]
2
(the index 2 refers to the fact that these are congruence classes
(mod 2)). The set
3
= 1
3
, ½2
3
g also has two
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Publisher Resources

ISBN: 9780123822178