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92
2. BASIC THEORY OF GROUPS
The row and column labeled by
e
are known. Show that the row labeled by
a
is determined by the requirement that each group element must appear
exactly once in each row and column; similarly, the column labeled by
a
is
determined. There are now four table entries left to determine. Show that
there are exactly two possible ways to complete the multiplication table
that are consistent with the constraints on multiplication tables. Show that
these two ways of completing the table yield the multiplication tables of
the two groups with four elements that we have already encountered.
2.1.12.
The group
˚.10/
of invertible elements in the ring
Z
10
has four
elements,
˚.10/
D f
Œ1Ł;Œ3Ł;Œ7Ł;Œ9Ł
g
. Is this group isomorphic to
Z
4
or to
the rotation group of the rectangle?
The group
˚.8/
also has four elements,
˚.8/
D f
Œ1Ł;Œ3Ł;Œ5Ł;Œ7Ł
g
. Is
this group isomorphic to
Z
4
or to the rotation group of the rectangle?
2.1.13.
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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

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