3Lagrange's Theorem

IA Groups



3.2 Left and right cosets
As
|aH|
=
|H|
and similarly
|H|
=
|Ha|
, left and right cosets have the same size.
Are they necessarily the same? We’ve previously shown that they might not be
the same. In some other cases, they are.
Example.
(i)
Take
G
= (
Z,
+) and
H
= 2
Z
. We have 0 + 2
Z
= 2
Z
+ 0 = even numbers
and 1 + 2
Z
= 2
Z
+ 1 = odd numbers. Since
G
is abelian,
aH
=
Ha
for all
a, G, H G.
(ii)
Let
G
=
D
6
=
hr, s | r
3
=
e
=
s
2
, rs
=
sr
1
i
. Let
U
=
hri
. Since
the cosets partition
G
, so one must be
U
and the other
sU
=
{s, sr
=
r
2
s, sr
2
= rs} = Us. So for all a G, aU = U a.
(iii)
Let
G
=
D
6
and take
H
=
hsi
. We have
H
=
{e, s}
,
rH
=
{r, rs
=
sr
1
}
and
r
2
H
=
{r
2
, r
s
}
; while
H
=
{e, s}, Hr
=
{r, sr}
and
Hr
2
=
{r
2
, sr
2
}
.
So the left and right subgroups do not coincide.
This distinction will become useful in the next chapter.