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2.1. FIRST RESULTS
91
2.1.7.
Suppose that
'
W
G
!
H
is an isomorphism of groups. Show that
G
is abelian if, and only if,
H
is abelian.
The following several exercises investigate groups with a small num
ber of elements by means of their multiplication tables. The requirements
ea
D
a
and
ae
D
a
for all
a
determine one row and one column of the
multiplication table. The other constraint on the multiplication table that
we know is that each row and each column must contain every group ele
ment exactly once. When the size of the group is small, these constraints
sufﬁce to determine the possible tables.
2.1.8.
Show that there is up to isomorphism only one group of order 2.
Hint:
Call the elements
f
e;a
g
. Show that there is only one possible mul
tiplication table. Since the row and the column labeled by
e
are known,
there is only one entry of the table that is not known. But that entry is de
termined by the requirement that each row and column contain each group
element.
2.1.9.
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 Fall '08
 EVERAGE
 Algebra, Multiplication

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