1
$\begingroup$

What are the smallest group with a derived series of length 2, 3 and 4?. I know that for n=2 the answer is S3 because that's the smallest metabelian non-abelian group. Could you help me out?

Thanks

$\endgroup$
  • $\begingroup$ Check small order known groups, say $$1\lhd V\lhd A_4\lhd S_4$$ $\endgroup$ – DonAntonio Nov 27 '13 at 5:52
  • $\begingroup$ Yes, that's a derived series of length four. Should I check that there aren't any groups that |G|<4! with a derived series of such length? $\endgroup$ – Pesi442 Nov 27 '13 at 5:57
2
$\begingroup$

For reference, the smallest groups of derived lengths 2, 3, 4, 5, and 6 respectively have orders 6, 24, 48, 432, and 1296 respectively.

The smallest group with derived length 2 is symmetric group on 3 points, $S_3$.

The smallest groups with derived length 3 are the symmetric group on 4 points, $S_4$, and the group of $2\times2$ matrices of determinant 1 with entries from a field with 3 elements, $\operatorname{SL}_2(\mathbb{Z}/3\mathbb{Z})$.

The smallest groups with derived length 4 are the group of $2\times2$ matrices of nonzero determinant with entries from a field with 3 elements, $\operatorname{GL}_2(\mathbb{Z}/3\mathbb{Z})$, and the so-called fake GL(2,3) from this answer.

The smallest group with derived length 5 is the group of affine transformations $$ \operatorname{AGL}_2(\mathbb{Z}/3\mathbb{Z}) = \left\{ \left[ \begin{smallmatrix} a & b & e \\ c & d & f \\ 0 & 0 & 1 \end{smallmatrix}\right] : a,b,c,d,e,f \in \mathbb{Z}/3\mathbb{Z}, ad-bc \neq 0 \right\} $$

The smallest groups with derived length 6 are three non-split extensions of $ \operatorname{AGL}_2(\mathbb{Z}/3\mathbb{Z})$ by a non-central, cyclic, normal subgroup of order 3. One of these is $\Gamma U_3(\mathbb{Z}/2\mathbb{Z})$, the split extension of $\operatorname{GL}_2(\mathbb{Z}/3\mathbb{Z})$ acting on an extraspecial group of order 27 and exponent 3; the other two are nonsplit.

An exploration of a similar problem (where instead of group order, the number of prime factors is consider, so 2, 4, 5, 7, and 8 respectively) is in Glasby (1986, 2005).

  • Glasby, S. P. “The composition and derived lengths of a soluble group.” J. Algebra 120 (1989), no. 2, 406–413. MR989905 DOI:10.1016/0021-8693(89)90204-4
  • Glasby, S. P. “Solvable groups with a given solvable length, and minimal composition length.” J. Group Theory 8 (2005), no. 3, 339–350. MR2137974 DOI:10.1515/jgth.2005.8.3.339
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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