16
Notes on JordanH¨older
Definition 16.1.
A
normal series
of a group
G
is a sequence of subgroups:
G
=
H
0
≥
H
1
≥
H
2
≥ · · · ≥
H
m
= 1
so that each subgroup is normal in the previous one (
H
i
E
H
i

1
). The quotient
groups
H
i

1
/H
i
is called a
subquotients
. A
refinement
of a normal series is
another normal series obtained by adding extra terms.
Example 16.2.
Here are some examples of normal series.
1.
S
n
> A
n
>
1 is a normal series for
G
=
S
n
for any
n
≥
1.
The
associated subquotients are
S
n
/A
n
∼
=
Z
/
2 and
A
n
/
1 =
A
n
.
2. For
n
= 4 this normal series has a refinement:
S
4
> A
4
> K >
1
The subquotients are
(a)
S
4
/A
4
∼
=
Z
/
2
(b)
A
4
/K
∼
=
Z
3
(Any group of order 3 is cyclic.)
(c)
K
∼
=
Z
/
2
×
Z
/
2
3. A normal series for the dihedral group
D
8
=
h
a, b

a
2
, b
4
, abab
i
is given
by
D
8
>
h
b
i
>
›
b
2
fi
>
1
.
All three subquotients are cyclic of order 2.
4.
G
=
GL
n
(
R
) has a normal series
GL
n
(
R
)
≥
SL
n
(
R
)
≥
Z
(
GL
n
(
R
))
≥
1
The subquotients are:
(a)
GL
n
(
R
)
/SL
n
(
R
)
∼
=
R
×
(b)
SL
n
(
R
)
/Z
(
GL
n
(
R
)) =
PSL
n
(
R
) by definition of the
projective
unimodular group
PSL
n
(
R
).
(c)
Z
(
GL
n
(
R
))
∼
=
R
×
since only scalar multiples of the identity are
central.
5. (A stupid example) Any group
G
has this normal series:
G
B
1
.
If
G
is a simple group, there is the only normal series for
G
.
1
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Definition 16.3.
A
composition series
is a normal series
G
=
G
0
> G
1
>
· · ·
> G
n
= 1
so that
G
i
is a maximal proper normal subgroup of
G
i

1
for 1
≤
i
≤
n
. The
subquotients
G
i

1
/G
i
are simple groups called the
composition factors
of
G
.
In Example 16.2, (1), for
n
≥
5 and (3) are composition series but (2) is
not because
K
is not simple. A composition series for
S
4
is given by refining
(3) by adding one more term:
H
= the cyclic group generated by (12)(34):
S
4
> A
4
> K > H >
1
The composition factors are
Z
/
2
,
Z
/
3
,
Z
/
2
,
Z
/
2. Example (4)
GL
n
(
R
) does
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 Spring '12
 ankit
 Algebra, Group Theory, Normal subgroup, Quotient group, Simple group, Normal Series

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