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2.5. COSETS AND LAGRANGE’S THEOREM
125
2.5.5.
What is the analogue of Proposition
2.5.3
, with left cosets replaced
with right cosets?
2.5.6.
Let
H
be a subgroup of a group
G
. Show that
aH
7!
Ha
±
1
deﬁnes
a bijection between left cosets of
H
in
G
and right cosets of
H
in
G
. (The
index of a subgroup was deﬁned in terms of left cosets, but this observation
shows that we get the same notion using right cosets instead.)
2.5.7.
For a subgroup
N
of a group
G
, prove that the following are equiv
alent:
(a)
N
is normal.
(b)
Each left coset of
N
is also a right coset. That is, for each
a
2
G
,
there is a
b
2
G
such that
aN
D
Nb
.
(c)
For each
a
2
G
,
aN
D
Na
.
2.5.8.
Suppose
N
is a subgroup of a group
G
and
ŒG
W
NŁ
D
2
. Show that
N
is normal using the criterion of the previous exercise.
2.5.9.
Show that if
G
is a ﬁnite group and
N
is a subgroup of index 2, then
for elements
a
and
b
of
G
, the product
ab
is an element of
N
if, and only
if, either both of
a
and
b
are in
N
or neither of
a
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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, Sets

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