Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $G$ be the group given by the set of invertible matrices of the form \begin{bmatrix}a & b & c\\0 & d & e\\0 & 0 & f\end{bmatrix} with $a,b,c,d,e,f \in \mathbb Z_3$.

Find the composition length of $G$ and its composition factors in terms of known groups, specifying which groups occur as composition factors and how many times each occurs in the composition series.

Attempt: I also know that the subset $N$ of $G$ of matrices where $a=d=f=1$ along the diagonal is a normal subgroup of $G$, that the centre of $N$, $Z(N)$ is \begin{bmatrix}1 & 0 & c\\0 & 1 & 0\\0 & 0 & 1\end{bmatrix} and that $G$ is soluble. Can I use this information to answer the question?

share|improve this question

1 Answer 1

Hint: By a direct computation you can show that $Z(N)$ is also the commutator subgroup of $N$. In particular, $N/Z$ is abelian, so you are reduced to studying $G/N$. To study this, look for a map $G \to D$, where $D$ is the group of diagonal matrices. (If "composition series" means the quotients occurring must be cyclic, and not just abelian, you will also need to study $D$ and $N/Z$ a little more closely.)

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.