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 Jan16 comment $|G|>2$ implies $G$ has non trivial automorphism @KevinCarlson, it might not be a direct product. For instance, it might be a direct sum... Jan16 comment $|G|>2$ implies $G$ has non trivial automorphism @tmastny, the $g_i$ are a basis for $G$, not an enumeration of $G$. So, by definition, one computes the action of $T$ on an element $g\in G$ by writing $g$ as a product of $g_i$'s and seeing what happens. In particular, in your example, $T(g_1g_i)=T(g_1)T(g_i)=g_2g_i$. Dec8 awarded Caucus Oct30 comment complete compact open topology Complete in what sense? en.wikipedia.org/wiki/Complete_topological_space Oct10 comment Fundamental group of a closed hyperbolic surface is Gromov hyperbolic This question is not research-level - any introductory text will contain a proof of this fact. Voting to close. Oct3 awarded Yearling Jun15 comment $|G|>2$ implies $G$ has non trivial automorphism @SaaqibMahmuud; well, suppose that $G$ is abelian and $2g=0$ for all $g$. Then you can check very easily that your group $G$ satisfies the axioms of a vector space over the field with two elements (which I denoted by $\mathbb{Z}/2$ in my answer). Then you argue exactly as I described above. If there are any further steps you have difficulty with, it would help if you said which ones they are! Apr10 comment Free Groups and Automatic Structures Also, let me add that since Mathoverflow (where this question was first posted) is by definition concerned with questions about research-level mathematics, I would (friendlily!) encourage non-mathematicians to start by posting their question on math.SE. Of course, if your question doesn't get an answer there, then trying MO is entirely appropriate. Apr10 comment Free Groups and Automatic Structures In that case I would encourage you to include the reading that you have done in the question. If you state that you've read 'Word processing in groups', say, but you're still confused about something, and can explain what that thing is, then you're more likely to get a helpful answer. Apr9 comment Free Groups and Automatic Structures Derek's answer is great, but this is surely contained in the standard book on the subject (eg Word Processing in Groups). I'm voting to close - MO is not a lazy alternative to reading the literature. Mar6 comment Fundamental group of row of spheres Please, let's not post answers to questions which are clearly off-topic. Feb25 revised Finding subgroups of a free group with a specific index Corrected a typo in Hall's formula. Feb3 comment Is there any connected n-manifold such that $H_n(X,Z)=Z\times Z$? @lanse2pty, any closed $n$-submanifold is clopen. Nov21 comment Covering Manifolds I always think of 'the Disc Theorem' as the conjunction of the Loop Theorem and Dehn's Lemma. Nov21 comment Covering Manifolds a) is obvious (ie not MO-level). b) is a consequence of the loop theorem, as already explained in the comments. Nov17 comment Product in an non-commutative group By the way, you might be interested to notice that any group which does have your property is commutative, because $ghg^{-1}h^{-1}=gg^{-1}hh^{-1}=1$ for any $g,h$. Nov17 answered Product in an non-commutative group Oct31 comment Given probability of event (Note: the above comment was written when this question and answer were posted on mathoverflow.) Oct31 comment Given probability of event Please don't answer questions that are not at research-level. They're off topic at MO. They can be migrated to math.stackexchange, where they are on topic and can be answered appropriately. Oct21 comment Universal Covering Space @t.b. , your idea is not the right way to construct the universal cover, but it is useful and interesting! If $X$ is a graph, say, then your limit $\varprojlim X_n$ is a profinite tree, and the profinite fundamental group of $X$, which is a free profinite group, acts on it. Ribes and Zalesskii have developed a very nice theory of profinite trees.