5.6. ADDITIONAL EXERCISES FOR CHAPTER 5
261
(c)
h
.1 2 3 4 5/;.2 3 5 4/
i Š
Z
4
Ë
Z
5
(and its ﬁve conjugates)
(d)
h
.1 2 3 4 5/;.2 5/.3 4/
i Š
D
5
(and its ﬁve conjugates)
(e)
h
.1 2 3 4 5/
i Š
Z
5
(and its ﬁve conjugates)
Remark 5.5.4.
There are 16 conjugacy classes of transitive subgroups of
S
6
, and seven of
S
7
. (See J. D. Dixon and B. Mortimer,
Permutation
Groups
, Springer–Verlag, 1996, pp. 58–64.) Transitive subgroups of
S
n
at least for
n
±
11
have been classiﬁed. Consult Dixon and Mortimer for
further details.
5.6. Additional Exercises for Chapter 5
5.6.1.
Let
G
be a ﬁnite group and let
H
be a subgroup. Let
Y
denote the
set of conjugates of
H
in
G
,
Y
D f
gHg
±
1
W
g
2
G
g
. As usual,
G=H
denotes the set of left cosets of
H
in
G
,
G=H
D f
gH
W
g
2
G
g
.
(a)
Show that
#
.G=H/
#
Y
D
ŒN
G
.H/
W
HŁ
.
(b)
Consider the map from
G=H
to
Y
deﬁned by
gH
7!
gHg
±
1
.
Show that this map is well deﬁned and surjective.
(c)
Show that the map in part (b) is one to one if, and only if,
H
D
N
G
.H/
. Show that, in general, the map is
ŒN
G
.H/
W
HŁ
to one
(i.e., the preimage of each element of
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 Fall '08
 EVERAGE
 Algebra, Addition, #, 1 W, Coset, ŒNG .H, J. D. Dixon

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