- 7.8 Nilpotent groups
- 7.9 Semidirect products
- 7.10 Classification of groups of small order

- Some group multiplication tables

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**7.8.1 Definition.**
For a group G we define the
** ascending central series**

Z_{1}(G)
Z_{2}(G)
. . .
of G as follows:

Z_{1}(G) is the center Z(G) of G;

Z_{2}(G)
is the unique subgroup of G with

Z_{1}(G)
Z_{2}(G) and
Z_{2}(G) /
Z_{1}(G) =
Z(G/Z_{1}(G)).

We define Z_{i}(G) inductively,
in the same way.

The group G is called
** nilpotent**
if there exists a positive integer n with
Z_{n}(G)=G.

**7.8.2 Proposition.**
If G_{1},
G_{2}, . . . ,
G_{n} are nilpotent groups,
then so is

G = G_{1}
×
G_{2}
×
. . .
×
G_{n}.

**7.8.3 Lemma.**
If P is a Sylow p-subgroup of a finite group G,
then the normalizer N(P) is equal to its own normalizer in G.

**7.8.4 Theorem.**
The following conditions are equivalent
for any finite group G.

**(1)**
G is nilpotent;

**(2)**
no proper subgroup H of G is equal to its normalizer N(H);

**(3)**
every Sylow subgroup of G is normal;

**(4)**
G is a direct product of its Sylow subgroups.

**7.8.5 Corollary.**
Let G be a finite nilpotent group of order n.
If m is any divisor of n,
then G has a subgroup of order m.

**7.8.6 Lemma. [Frattini's Argument]**
Let G be a finite group,
and let H be a normal subgroup of G.
If P is any Sylow subgroup of H, then G=HN(P),
and [G:H] is a divisor of |N(P)|.

**7.8.7 Proposition.**
A finite group is nilpotent if and only if
every maximal subgroup is normal.

Then G is called the

**Example.** 7.9.6.
(D_{n} is a semidirect product.)

Consider the dihedral group D_{n},
described by generators a of order n and b of order 2,
with the relation ba=a^{-1}b.
Then <a> is a normal subgroup,
<a><b>={e}, and
<a><b>=D_{n}.
Thus the dihedral group is a semidirect product of
a cyclic subgroups of order n by a cyclic subgroup of order 2.

Definition 7.9.1 describes an ``internal'' semidirect product. We now use the automorphism group to give a general definition of an ``external'' semidirect product.

**7.9.2 Definition.**
Let G be a multiplicative group,
and let X be an abelian group, denoted additively.
Let :G->Aut(X)
be a group homomorphism. The
** semidirect product of X and G relative to
** is

X
_{}
G
= { (x,a) | x X,
a G }

with the operation
(x_{1},
a_{1})
(x_{2},
a_{2}) =
( x_{1} +
(a_{1})
[x_{2}],
a_{1}
a_{2}),
for x_{1},
x_{2}
X and
a_{1},
a_{2}
G.

**7.9.3 Proposition.**
Let G be a multiplicative group, let X be an additive group, and let
:G->Aut(X)
be a group homomorphism.

**(a)**
The semidirect product
X
_{}
G is a group.

**(b)**
The set { (x,a)
X
_{}
G | x = 0 }
is a subgroup of
X
_{}
G
that is isomorphic to G.

**(c)**
The set N = { (x,a) X
_{}
G | a = e }
is a normal subgroup of
X _{}G
that is isomorphic to X, and
(X _{}G)/N is isomorphic to G.

We can now give a more general definition.
We say that
**Z**_{n}
_{}
**Z**_{n}^{x} is the
** holomorph**
of **Z**_{n},
denoted by H_{n}.

**7.9.4 Definition.**
Let G be a group and let X be an abelian group.
If G acts on X and a(x+y)=ax+ay,
for all aG and
x,yX,
then we say that G
** acts linearly**
on X.

**7.9.5 Proposition.**
Let G be a group and let X be an abelian group.
Then any group homomorphism from G into the group Aut(X) of
all automorphisms of X defines a linear action of G on X.
Conversely, every linear action of G on X arises in this way.

**7.9.6 Proposition.**
Let G be a multiplicative group with a normal subgroup N,
and assume that N is abelian.
Let :G->G/N
be the natural projection.
The following conditions are equivalent:

**(1)**
There exists a subgroup K of G such that
NK={e} and NK=G;

**(2)**
There exists a homomorphism
:G/N->G
such that
is the identity on G/N;

**(3)**
There exists a homomorphism
:G/N->Aut(N)
such that
N
_{}
(G/N)G.

**7.10.1 Proposition.**
Any nonabelian group of order 6 is isomorphic to
S_{3}.

**7.10.2 Proposition.**
Any nonabelian group of order 8
is isomorphic either to D_{4} or
to the quaternion group Q.

**7.10.3 Proposition.**
Let G be a finite group.

**(a)**
Let N be a normal subgroup of G.
If there exists a subgroup H such that
HN={e} and |H|=[G:N], then
GN H.

**(b)**
Let G be a group with
|G|=p^{n}q^{m},
for primes p,q.
If G has a unique Sylow p-subgroup P,
and Q is any Sylow q-subgroup of G, then
GP Q.
Furthermore, if Q' is any other Sylow q-subgroup, then
P Q' is isomorphic to
P Q.

**(c)**
Let G be a group with |G|=p^{2}q,
for primes p,q.
Then G is isomorphic to a semidirect product of its Sylow subgroups.

**7.10.4 Lemma.**
Let G,X be groups, let
,
:G->Aut(X),
and let
,
be the corresponding linear actions of G on X.

Then
X
_{}
G
X
_{}
G
if there exists
Aut(G)
such that
=
.

**7.10.5 Proposition.**
Any nonabelian group of order 12 is isomorphic to one of

A_{4}, D_{6}, or
**Z**_{3}
**Z**_{4}.

The following table summarizes the above information
on the isomorphism classes of groups of order less than sixteen.

Order 2: **Z**_{2}

Order 3: **Z**_{3}

Order 4: **Z**_{4},
**Z**_{2}
×
**Z**_{2}

Order 5: **Z**_{5}

Order 6: **Z**_{6},
S_{3}

Order 7: **Z**_{7}

Order 8: **Z**_{8},
**Z**_{4}
×
**Z**_{2},
**Z**_{2}
×
**Z**_{2}
×
**Z**_{2},
D_{4}, Q

Order 9: **Z**_{9},
**Z**_{3}
×
**Z**_{3}

Order 10: **Z**_{10},
D_{5}

Order 11: **Z**_{11}

Order 12: **Z**_{12},
**Z**_{6}
×
**Z**_{2}
A_{4}, D_{6},
**Z**_{3}
**Z**_{4}

Order 13: **Z**_{13}

Order 14: **Z**_{14},
D_{7}

Order 15: **Z**_{15}

**7.10.6 Proposition.**
Let G be a finite simple group of order n,
and let H be any proper, nontrivial subgroup of G.

**(a)**
If k = [G:H], then n is a divisor of k!.

**(b)**
If H has m conjugates, then n is a divisor of m!.

**7.10.7 Proposition.**
The alternating group
A_{5}
is the smallest nonabelian simple group.