I am trying to find all groups of order $39.$ So far, I have shown that only two such groups exist ($\mathbb{Z}_{39}$ and one nonabelian group). My question is: how can I find a presentation for the nonabelian group? I know that it contains elements of order $3$ and $13$, so I need two generators. But I also need to find some relation that exists between the two generators. Is there any systematic way to go about doing this?
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The 13-sylow, assume generated by $x$, is normal in $G$. Let $y$ be a generator for a 3-sylow. Then by normality, we have $$yxy^{-1} = x^i$$ for some integer $i$. Hence $i^3 \equiv 1 \pmod{13}$, which implies $i\equiv 1,3,9 \pmod{13}$.
The case $i=1$ gives the cyclic group $C_3 \times C_{13}$. The groups given by $i=3,9$ are isomorphic, so a presentation is given by $$\{x,y\mid x^{13} = 1, y^3 = 1, yx= x^3y \}$$
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4$\begingroup$ @Analytical $$yxy^{-1} = x^i \implies y^2xy^{-2} = yx^iy^{-1} = x^{i^2} \implies x=y^3xy^{-3} = yx^{i^2}y^{-1} = x^{i^3} $$ $\endgroup$– piscoSep 27, 2017 at 1:59
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$\begingroup$ I understand, but why do you then need it to be congruent to $1 \pmod{13}$? $\endgroup$ Sep 27, 2017 at 2:05
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1$\begingroup$ Becuase $x=x^{i^3}$, and $x$ has order 13, this implies $i^3 \equiv 1 \pmod{13}$. $\endgroup$– piscoSep 27, 2017 at 2:07
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3$\begingroup$ If $yxy^{-1} = x^3$, then $y^2 x y^{-1} = x^9 $, this tells us the case $i = 9$ arises from the case $i = 3$ via replacing $y$ by $y^2$. But the subgroups generates by $y$ and $y^2$ are just the same. $\endgroup$– piscoSep 27, 2017 at 2:11