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 Apr 20 comment Is the Cayley graph of a word-hyperbolic group a CAT(0) metric space? @studiosus what if we assume $G$ is torsion-free? Apr 15 comment Intersecting circles and the sine and cosine rules Yeah, using similar logic to finding $\sin\theta$ and $\tan\theta$ if you are given that $\cos\theta=12/13$ with $0\leq\theta\leq\pi$. (Then $\sin\theta=5/13$ and $\tan\theta=5/12$.) Apr 15 revised Intersecting circles and the sine and cosine rules edited tags Apr 15 comment Intersecting circles and the sine and cosine rules I do not understand why $pa=pb$ (and I think it is wrong in general). For example, using the easier $\sqrt2$, $2$ and $1+\sqrt3$ values I mentioned in the question gives $(pa)^2=8-2\sqrt3$ while $(pb)^2=4-\sqrt2$. Apr 14 comment Intersecting circles and the sine and cosine rules How do you find $\alpha$ though? This requires a calculator (in general). Apr 14 comment Intersecting circles and the sine and cosine rules @Mick How would you do that without using a half- or double-angle formulae? Apr 14 comment Intersecting circles and the sine and cosine rules @almagest Because that is too simple! Apr 14 asked Intersecting circles and the sine and cosine rules Apr 14 comment For which finite groups $G$ does a finite group $H$ exist, such that $Aut(H)$ is isomorphic to $G\$? (Incidentally, if instead of considering $\operatorname{Aut}(H)$ you consider $\operatorname{Out}(H):=\operatorname{Aut}(H)/\operatorname{Inn}(H)$ the problem becomes more tractable. For example, every finite group $G$ can be realised as the outer automorphism group $\operatorname{Out}(H)$ where $H$ is the fundamental group of a closed hyperbolic $3$-manifold.) Apr 14 revised For which finite groups $G$ does a finite group $H$ exist, such that $Aut(H)$ is isomorphic to $G\$? added 15 characters in body Apr 14 comment For which finite groups $G$ does a finite group $H$ exist, such that $Aut(H)$ is isomorphic to $G\$? You can distil my answer in lhf's link to prove that no cyclic group of odd order occurs as $\operatorname{Aut}(H)$ where $H$ is finite. (If $\operatorname{Aut}(H)$ is cyclic and non-trivial then $H$ is abelian, apply fundamental theorem for finitely generated abelian groups and notice that, as $\operatorname{Aut}(H)$ is non-trivial, there always exists an automorphism of order $2$.) Apr 13 comment What is the difference between a $p$-group and a Sylow group? The word "sub". Sylow subgroup. Apr 11 comment I don't understand what a “free group” is! "Well, $x_1*x_2$...(is a) new word $x_1x_2$. Continuing this way, we keep adding the newly created words and reach infinity." It is worth pointing out that all words have finite length. For example, you can have $x^n$ for all $n\in\mathbb{Z}$ (including $n=0$), but you cannot have $x^{\infty}$. Apr 10 comment Find the rank and the free generators @CaptainLama That's okay - the only reason I know which is the Cayley Complex and which is the Presentation Complex is that I kept getting them mixed up so made a conscious effort to not make this mistake anymore! Apr 10 comment Find the rank and the free generators Also see here. There is a nice "conceptual" version of the algorithm, which uses covering spaces. The issue with it is that is does not give you explicit generators. I am sure you can modify it to find these generators also, but I have never quite gathered up the motivation to do this. The algebraic version from Magnus, Karrass and Solitar gives you the generators, but is a "drier" algorithm. Apr 10 comment Find the rank and the free generators @CaptainLama The Cayley complex is the universal cover of the presentation complex, so it has trivial fundamental group. (Also...your italicised "the"...there are actually two classic books entitled "Combinatorial Group Theory"; Lyndon and Schupp named their book in honour of Magnus, Karrass and Solitar. From memory, contain Reidemeister-Schreirer, but if either contains a more geometric or conception version it will be Lyndon and Schupp.) Apr 10 comment Is the Cayley graph of a word-hyperbolic group a CAT(0) metric space? Perhaps the word "graph" should have been "complex". Refuting this is harder, although I am sure it is still false. (Even if all hyperbolic groups are $CAT(0)$, I would be surprised if it was because their Cayley Complexes' are $CAT(0)$.) Apr 10 comment Find the rank and the free generators The Reidermeister-Shrier algorithm is what you're after. (I'm almost certain I've misspelt it!) Apr 8 comment Examples of infinite Semi-direct products For your second point, I don't know off hand. $\mathbb{Z}$ works for two reasons - it has a copy $Z$ of itself in the preimage, and if $k\in K\cap Z$ is non-trivial then it "kills" part of $Z$ so $Z$ maps to a finite group, a contradiction. The second point is easy to mimick, e.g. a simple group or an almost finite group. I believe that the first point is the tough one to get working. Apr 8 revised Examples of infinite Semi-direct products added 4 characters in body