Meta cyclic p-group

Question

While studying meta cyclic p groups, I came across an interesting class of meta cyclic groups which can be written as semi-direct product of two cyclic p-groups of order $p^m$ and $p^n$ respectively. These kind of groups are called split meta cyclic p-groups. I am trying to write their presentation. Can someone help me present them in a nice way?

Answer 1

A split metacyclic group is split extension of a cyclic group by a cyclic group, a group $G$ for which the short exact sequence$$\displaystyle \{e\}\rightarrow K\xrightarrow{\text{f}} G\xrightarrow{g} H\rightarrow \{e\}$$ splits, i.e. given any epimorphism $ g : G \rightarrow H $,\ there exist a map $ h : H \rightarrow G $, such that composition of $g$ and $h$ is identity on $H$ . Using this, $G$ can be written as a semi-direct product of $K$ by $H$.

Any non-abelian split metacyclic-p-group has presentation either $$ <\ a, b\ : a^{2^m} = b^{2^n} = e, bab^{-1} = a^{-1+2^k}> $$ where $max\{2,m-n\} \leq k \leq m$ for $p=2$ or

$$ <\ a, b\ : a^{p^m} = b^{p^n} = e, bab^{-1} = a^{1+p^k}>$$ where $(m-n) \leq k < m$ with $k \geq 1, $ for all primes $p$.

Reference Article: Golasiński, Marek; Gonçalves, Daciberg Lima, On automorphisms of split metacyclic groups., Manuscr. Math. 128, No. 2, 251-273 (2009). ZBL1160.20017.