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5.4. GROUP ACTIONS AND GROUP STRUCTURE
257
and
NA
D
G
, because
j
NA
j D
j
N
jj
A
j
j
N
\
A
j
D
28
. Thus
G
is the semidirect
product of
N
and
A
.
The abelian groups of order 28 are
Z
7
±
Z
4
and
Z
7
±
Z
2
±
Z
2
. To
classify the nonabelian groups of order 28, we have to classify the non
trivial homomorphisms from groups of order 4 into Aut
.
Z
7
/
Š
Z
6
.
Aut
.
Z
7
/
has a unique subgroup of order 2, generated by the automor
phism
j
W
ŒxŁ
7
7!
Œ
²
xŁ
7
. Any nontrivial homomorphism from a group of
order 4 into Aut
.
Z
7
/
must have image
h
j
i
, since the size of the image is a
common divisor of 4 and 6. So we are looking for homomorphisms from
a group of order 4 onto
h
j
i Š
Z
2
.
Z
4
has a unique homomorphism
˛
onto
h
j
i
determined by
˛
W
Œ1Ł
4
7!
j
. Therefore, up to isomorphism,
Z
7
Ì
˛
Z
4
is the unique nonabelian group
of order 28 with
2
–Sylow subgroup isomorphic to
Z
4
. This group is gen
erated by elements
a
and
b
satisfying
a
7
D
b
4
D
1
and
bab
±
1
D
a
±
1
.
See Exercise
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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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