I am studying the article "INFINITE METACYCLIC GROUPS AND THEIR NON-ABELIAN TENSOR SQUARES". In this article authors have proved a lemma, (Lemma 3.1):
Lemma 3.1. Let $m,n, r$ be integers with $m,n$ non-negative, $\gcd(r,m)=1$, and let $$G = \langle a,b \mid a^m = b^n = 1, bab^{-1} = a^{r}\rangle,$$ a split metacyclic group. Then every $g\in G$ can be written as $g = a^\alpha b^\beta$, where $\alpha ,\beta$ are integers which are unique modulo $m$ and $n$, respectively. If $h \in G$ with $h = a^\gamma b^\delta$, then we have the following multiplication, conjugation formulae in $G$:
- $gh = a^{\alpha+\gamma r^{\delta}}b^{\beta+\delta}$
- $^g h$ = $a^{\gamma(1-r^\beta)+\alpha r^{\gamma}}b^{\beta}$
The authors gave a very short proof. When I am trying to prove the part (2) of the lemma I am getting some mistakes. Like take $g = a^\alpha b^\beta$ and $h=a$ or $h=b$, then I think the formula fails.
I want to prove the second part of the lemma, here $^gh=ghg^{-1}$. Please help me.
I agree, both equations are wrong - the authors have been careless. I make it $$gh = a^{\alpha +\gamma r^\beta}b^{\beta+\delta}$$ $$ghg^{-1} = a^{\alpha(1-r^{\delta})+ \gamma r^\beta} b^\delta.$$