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I am trying to find a composition series for $F_{20}$, the set of matrices of the form $\left[ \begin{array}{cc} m & n \\ 0 & 1 \end{array} \right]$ where $n\in \mathbb{Z}_5$ and $m\in \mathbb{Z}^{\times}_{5}$.

This is problem 7.6.5 of Beachy & Blair's algebra.

I can see that, for example, $\left[ \begin{array}{cc} \bar{2} & 0 \\ 0 & 1 \end{array} \right]$ generates a group of order 4, since $\mathbb{Z}^{\times}_{5}=\{\bar{1},\bar{2},\bar{3},\bar{4}\}$.

If I could find a subgroup of order 10, it would have index 2 and be normal in $F_{20}$, and its quotient group would clearly be simple so that I would be done.

Another part of the problem is to find a descending series of commutator subgroups of $F_{20}$, and I do not know how to begin.

Thank you very much for your help.

Harold J. M.
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Try $H = \left\{\begin{bmatrix}\pm1&a\\0&1\end{bmatrix}: a\in \Bbb Z_5\right\}$

David Wheeler
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