# Solvable Group

• Apr 12th 2010, 11:27 AM
mathman88
Solvable Group
Show $\displaystyle B=\left\{ \left. \left(\begin{array}{cc}x&z\\0&y\end{array}\right) \;\; \right| \;\; x,y\in \mathbb{F}_p^\times,\; z\in\mathbb{F}_p \right\}$ is a solvable group.

The group operation is matrix multiplication.
• Apr 12th 2010, 11:38 AM
tonio
Quote:

Originally Posted by mathman88
Show $\displaystyle B=\left\{ \left. \left(\begin{array}{cc}x&z\\0&y\end{array}\right) \;\; \right| \;\; x,y\in \mathbb{F}_p^\times,\; z\in\mathbb{F}_p \right\}$ is a solvable group.

The group operation is matrix multiplication.

Check the commutator group: it then should be "almost" obvious that it is abelian so...

Tonio
• Apr 12th 2010, 11:47 AM
mathman88
Quote:

Originally Posted by tonio
Check the commutator group: it then should be "almost" obvious that it is abelian so...

Tonio

$\displaystyle [B,B] = \left\{ \left. \left(\begin{array}{cc}1&z\\0&1\end{array}\right) \;\; \right| \;\; z\in\mathbb{F}_p \right\}$

And it easy to see $\displaystyle [B,B]$ is abelian.

We then have $\displaystyle \{1\} \triangleleft [B,B] \triangleleft B$ and $\displaystyle [B,B]/\{1\} = [B,B]$ is cyclic, but how is $\displaystyle B/[B,B]$ cyclic?
• Apr 12th 2010, 11:58 AM
tonio
Quote:

Originally Posted by mathman88
$\displaystyle [B,B] = \left\{ \left. \left(\begin{array}{cc}1&z\\0&1\end{array}\right) \;\; \right| \;\; z\in\mathbb{F}_p \right\}$

And it easy to see $\displaystyle [B,B]$ is abelian.

How does this show $\displaystyle B$ is solvable?

(!!) Because then the derived series is finite, of course: $\displaystyle G\supseteq G'\supseteq G''=1$ and this an abelian series for the group...what definition of "solvable" do you have?

Tonio
• Apr 12th 2010, 12:02 PM
mathman88
Quote:

Originally Posted by tonio
(!!) Because then the derived series is finite, of course: $\displaystyle G\supseteq G'\supseteq G''=1$ and this an abelian series for the group...what definition of "solvable" do you have?

Tonio

$\displaystyle G$ is solvable $\displaystyle \iff \{1\} \triangleleft G_i \triangleleft \ldots G_1 \triangleleft G_0=G$ and $\displaystyle \forall \; j, \; G_j/G_{j+1}$ is cyclic.
• Apr 12th 2010, 12:14 PM
tonio
Quote:

Originally Posted by mathman88
$\displaystyle G$ is solvable $\displaystyle \iff \{1\} \triangleleft G_i \triangleleft \ldots G_1 \triangleleft G_0=G$ and $\displaystyle \forall \; j, \; G_j/G_{j+1}$ is cyclic.

This is not the usual definition, and it's a rather misleading and even incorrect one (imo, of course. See following note), but it never minds: since the group $\displaystyle B$ is finitely generated then $\displaystyle B\slash [B,B]$ is finitely generated and thus can be decomposed in a direct product of cyclic groups...

Note: for your definition to work it MUST be that G is fin. generated, or at least that all its abelian factor groups are, otherwise it fails: an infinitely generated abelian group wouldn't be solvable according to your definition which, of course, is absurd.

Tonio