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Say we have a finite group $G$ generated by $\langle g_1,g_2,\ldots ,g_n\rangle$ which is a minimal generating set. What information is required from this set in order to determine the whole group?

For cyclic groups, all that is needed is the order of some $g_1$, but for a set generated by two elements, I imagine you need the order of $g_1$, $g_2$, and the order of some combination of the two elements.

The motivation is to better understand Cayley graphs and potentially use these ideas to better understand group extensions.

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    $\begingroup$ well, the relations. Every group has a presentation by generators and relations and that representation determines the group up to isomorphism. For instance the groups $\mathbb{Z}_{m}$ are of the form $\langle x \mid x^{m} = e\rangle$, and the dihedral group is of the form $D_{n} = \langle \sigma , \tau \mid \sigma^{n} = \tau^{2} = e , \tau \sigma \tau^{-1} = \sigma^{-1} \rangle$ $\endgroup$ Commented Aug 22 at 9:22
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    $\begingroup$ The key words are "presentations" and "relators"; relators describe loops in the Cayley graph, and are not just about orders of elements. Your question is a bit broad and doesn't admit a precise answer (other than maybe "read a book", e.g. Magnus, Karrass and Solitar's Combinatorial Group Theory, or maybe Rotman's book but I've never actually read that one properly). However, I wrote an answer here that puts presentations into the context of group extensions. $\endgroup$ Commented Aug 22 at 10:02

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The key words are "presentation" and "relator"; relators describe loops in the Cayley graph, and are not just about orders of elements. Group presentations are systems of generators and relators. Another key word is "free group"; every group is isomorphic to a quotient of a free group, and presentations formally define quotients of a free group.

The first presentations usually encountered are dihedral groups: $$ D_{2n}\cong\langle a, b\mid b^{-1}aba, a^n, b^2\rangle $$ Here, $\{a, b\}$ is my generating set, and $\{b^{-1}aba, a^n, b^2\}$ is my set of relators. A relator $W$ is a word over the generators and their inverses, and has the property that $W=1$ in the group. Note that the relator "$b^{-1}aba$" is not describing the order of an element, but instead says that $b^{-1}ab=a^{-1}$, i.e. that conjugation by the element $b$ inverts the element $a$. Another example of a group presentation would be $$ \mathbb{Z}_n\times\mathbb{Z}_2\cong\langle a, b\mid b^{-1}aba^{-1}, a^n, b^2\rangle $$ This is identical to the above presentation, except the relator "$b^{-1}aba$" has now been changed to "$b^{-1}aba^{-1}$". This is equivalent to $ab=ba$, and means that the elements $a$ and $b$ commute.

I wrote an answer here that puts presentations into the context of group extensions.

Presentations look very attractive, but they are difficult to work with. For example, it is in general undecidable (i.e. there is no general algorithm which will determine) if a given presentation defines a finite group. The above link to presentations in the context of group extensions discusses a similar problem: if a presentation looks like a group extension, then when can we conclude that it is indeed a group extension of the given groups? Furthermore, the "$\cong$" symbols I wrote above are correct, but hard to justify without further theory.

If you are interested in this topic, a good first step would be to read Magnus, Karrass and Solitar's book Combinatorial Group Theory, which has excellent exercises, or maybe Rotman's book, which is more friendly and modern, although I have not read it properly so can only half-recommend it.

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    $\begingroup$ Just a note on terminology: I believe that equations like $b^{-1}ab=a^{-1}$ are normally referred to as relations rather than relators. A relator is a word that evaluates to the identity in the group. So, a relator correspohding to the above relation is $b^{-1}aba$. $\endgroup$ Commented Aug 25 at 13:23
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    $\begingroup$ @DerekHolt Thanks - I agree and I'm not sure why I didn't catch this! I've edited the answer to just be about relators. $\endgroup$ Commented Aug 25 at 14:06
  • $\begingroup$ Final question. If we are given a group $E$ with normal subgroup $A$ with presentation $\langle \mathbf{x}\mid\mathbf{r}\rangle$ and the corresponding quotient group $Q$ with presentation $\langle \mathbf{y}\mid\mathbf{s}\rangle$, where both presentations are “minimal” (removing any generator or relator makes the presentation incomplete), is $ \langle \mathbf{x, y}\mid SW_S^{-1} (S\in\mathbf{s}), \mathbf{r}, \mathbf{t}\rangle $ a “minimal” presentation of $E$? In terms of Cayley graphs, I imagine this to be true since cayley graphs are connected iff the generating set is sufficient. $\endgroup$ Commented Aug 28 at 4:21
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    $\begingroup$ @NullSimplex The problem of determining if a presentation is minimal is difficult in general. Your point about connectedness is just about generators, but you also need to include relators. I do not know any counter-examples off hand, but I would be surprised if this result was true. You should ask it as a separate question. $\endgroup$ Commented Aug 28 at 16:30

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