Editing Solvable group (section)

== Examples ==

=== Abelian groups ===
The basic example of solvable groups are abelian groups. They are trivially solvable since a subnormal series is formed by just the group itself and the trivial group. But non-abelian groups may or may not be solvable.

=== Nilpotent groups ===
More generally, all [[nilpotent group]]s are solvable. In particular, finite [[p-group|''p''-groups]] are solvable, as all finite [[p-group|''p''-groups]] are nilpotent.

==== Quaternion groups ====
In particular, the [[quaternion group]] is a solvable group given by the group extension<blockquote><math>1 \to \mathbb{Z}/2 \to Q \to \mathbb{Z}/2 \times \mathbb{Z}/2 \to 1</math></blockquote>where the kernel <math>\mathbb{Z}/2</math> is the subgroup generated by <math>-1</math>.

=== Group extensions ===
[[Group extension]]s form the prototypical examples of solvable groups. That is, if <math>G</math> and <math>G'</math> are solvable groups, then any extension<blockquote><math>1 \to G \to G'' \to G' \to 1</math></blockquote>defines a solvable group <math>G''</math>. In fact, all solvable groups can be formed from such group extensions.

=== Non-abelian group which is non-nilpotent ===
A small example of a solvable, non-nilpotent group is the [[symmetric group]] ''S''<sub>3</sub>. In fact, as the smallest simple non-abelian group is ''A''<sub>5</sub>, (the [[alternating group]] of degree 5) it follows that ''every'' group with order less than 60 is solvable.

=== Finite groups of odd order ===
The [[Feit–Thompson theorem]] states that every finite group of odd order is solvable. In particular this implies that if a finite group is simple, it is either a prime cyclic or of even order.

=== Non-example ===
The group ''S''<sub>5</sub> is not solvable &mdash; it has a composition series {E, ''A''<sub>5</sub>, ''S''<sub>5</sub>} (and the [[Jordan–Hölder theorem]] states that every other composition series is equivalent to that one), giving factor groups isomorphic to ''A''<sub>5</sub> and ''C''<sub>2</sub>; and ''A''<sub>5</sub> is not abelian. Generalizing this argument, coupled with the fact that ''A''<sub>''n''</sub> is a normal, maximal, non-abelian simple subgroup of ''S''<sub>''n''</sub> for ''n'' > 4, we see that ''S''<sub>''n''</sub> is not solvable for ''n'' > 4. This is a key step in the proof that for every ''n'' > 4 there are [[polynomial]]s of degree ''n'' which are not solvable by radicals ([[Abel–Ruffini theorem]]). This property is also used in complexity theory in the proof of [[NC (complexity)|Barrington's theorem]].

=== Subgroups of GL<sub>2</sub> ===
Consider the subgroups<blockquote><math>B = \left\{ \begin{bmatrix}
* & * \\
0 & *
\end{bmatrix} \right\} \text{, }
U = \left\{ \begin{bmatrix}
1 & * \\
0 & 1
\end{bmatrix} \right\}</math> of <math>GL_2(\mathbb{F})</math></blockquote>for some field <math>\mathbb{F}</math>. Then, the group quotient <math>B/U</math> can be found by taking arbitrary elements in <math>B,U</math>, multiplying them together, and figuring out what structure this gives. So<blockquote><math>\begin{bmatrix}
a & b \\
0 & c 
\end{bmatrix} \cdot
\begin{bmatrix}
1 & d \\
0 & 1 
\end{bmatrix} =
\begin{bmatrix}
a & ad + b \\
0 & c
\end{bmatrix}

</math></blockquote>Note the determinant condition on <math>GL_2

</math> implies <math>ac \neq 0

</math>, hence <math>\mathbb{F}^\times \times \mathbb{F}^\times \subset B

</math> is a subgroup (which are the matrices where <math>b=0

</math>). For fixed <math>a,b

</math>, the linear equation <math>ad + b = 0

</math> implies <math>d = -b/a

</math>, which is an arbitrary element in <math>\mathbb{F}

</math> since <math>b \in \mathbb{F}

</math>. Since we can take any matrix in <math>B

</math> and multiply it by the matrix<blockquote><math>\begin{bmatrix}
1 & d \\
0 & 1
\end{bmatrix}

</math></blockquote>with <math>d = -b/a

</math>, we can get a diagonal matrix in <math>B

</math>. This shows the quotient group <math>B/U \cong \mathbb{F}^\times \times \mathbb{F}^\times</math>.

==== Remark ====
Notice that this description gives the decomposition of <math>B

</math> as <math>\mathbb{F} \rtimes (\mathbb{F}^\times \times \mathbb{F}^\times)

</math> where <math>(a,c)

</math> acts on <math>b

</math> by <math>(a,c)(b) = ab

</math>. This implies <math>(a,c)(b + b') = (a,c)(b) + (a,c)(b') = ab + ab'

</math>. Also, a matrix of the form<blockquote><math>\begin{bmatrix}
a & b \\
0 & c
\end{bmatrix}</math></blockquote>corresponds to the element <math>(b) \times (a,c)</math> in the group.

=== Borel subgroups ===
For a [[linear algebraic group]] <math>G</math>, a [[Borel subgroup]] is defined as a subgroup which is closed, connected, and solvable in <math>G</math>, and is a maximal possible subgroup with these properties (note the first two are topological properties). For example, in <math>GL_n</math> and <math>SL_n</math> the groups of upper-triangular, or lower-triangular matrices are two of the Borel subgroups. The example given above, the subgroup <math>B</math> in <math>GL_2</math>, is a Borel subgroup.

==== Borel subgroup in GL<sub>3</sub> ====
In <math>GL_3</math> there are the subgroups<blockquote><math>B = \left\{
\begin{bmatrix}
* & * & * \\
0 & * & * \\
0 & 0 & *
\end{bmatrix}
\right\}, \text{ }
U_1 = \left\{
\begin{bmatrix}
1 & * & * \\
0 & 1 & * \\
0 & 0 & 1
\end{bmatrix}
\right\}</math></blockquote>Notice <math>B/U_1 \cong \mathbb{F}^\times \times \mathbb{F}^\times \times \mathbb{F}^\times</math>, hence the Borel group has the form<blockquote><math>U\rtimes
(\mathbb{F}^\times \times \mathbb{F}^\times \times \mathbb{F}^\times)

</math></blockquote>

==== Borel subgroup in product of simple linear algebraic groups ====
In the product group <math>GL_n \times GL_m</math> the Borel subgroup can be represented by matrices of the form<blockquote><math>\begin{bmatrix}
T & 0 \\
0 & S
\end{bmatrix}</math></blockquote>where <math>T</math> is an <math>n\times n</math> upper triangular matrix and <math>S</math> is a <math>m\times m</math> upper triangular matrix.

=== Z-groups ===
Any finite group whose [[Sylow group|''p''-Sylow subgroups]] are cyclic is a [[semidirect product]] of two cyclic groups, in particular solvable.  Such groups are called [[Z-group]]s.