Timeline for Some counter examples in group theory
Current License: CC BY-SA 3.0
17 events
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| Sep 19, 2015 at 11:57 | comment | added | user91132 | user1729: $\mathbb{Z}/2\mathbb{Z}$ is an abelian group with no automorphism of even order. But of course its automorphism group is not isomorphic to $\mathbb{Z}$ either. Your argument shows that $\Aut(G)$ cannot be an abelian non-trivial group with no elements of order two. So for example any non-trivial finite abelian group of odd order is ruled out. | |
| Sep 11, 2015 at 13:32 | comment | added | user1729 | For (5), $\mathbb{Z}$ works. I think also any cyclic group of odd (prime?) order. The idea is to note that $G/Z(G)$ embeds into $\operatorname{Aut}(G)$, and hence $G$ is abelian (standard undergrad exercise). Such a group always has an automorphism of even order, assuming AOC (less standard exercise, not sure of level), a contradiction. | |
| Sep 7, 2015 at 8:39 | review | Close votes | |||
| Sep 10, 2015 at 13:48 | |||||
| Sep 5, 2015 at 10:08 | comment | added | Ali Taghavi | @JasonStarr Thank you for your comment. As you can guess the "s" character in my keyboard had some problem and I was not aware of that. Any way, is there an obtruction to have compact Haudorff space X with Homeo(X) isomorphic to G? | |
| Sep 4, 2015 at 23:26 | comment | added | Jason Starr | @AliTaghavi: "could you pleae more explain on thi topology?" I would like to recommend that you use the spell checker that is built into the MO comments system. A subset $U$ of $X$ is open with respect to the topology I mention if and only if it satisfies all of the following conditions: for every $(g,h)$ in $G\times G$, (a) if $U$ contains $t_g$ then $U$ contains $u_{g,h}$, (b) if $U$ contains $t_h$ or $u_{g,h}$, then $U$ contains $v_{g,h}$, and (c) if $U$ contains $t_{g\cdot h}$ or $v_{g,h}$ then $U$ contains $w_{g,h}$. | |
| Sep 4, 2015 at 21:34 | history | edited | Ali Taghavi |
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| Sep 4, 2015 at 21:00 | comment | added | Ali Taghavi | @JasonStarr could you pleae more explain on thi topology?any reference?I there obstruction to find a compact hausdorff space? | |
| Sep 4, 2015 at 20:57 | comment | added | Ali Taghavi | @YiftachBarnea thank you for your comment. i revise the question. | |
| Sep 4, 2015 at 20:56 | history | edited | Ali Taghavi | CC BY-SA 3.0 |
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| Sep 4, 2015 at 19:08 | comment | added | Yiftach Barnea | Do you mean any structure or a structure? | |
| Sep 4, 2015 at 18:57 | comment | added | Jason Starr | Every group $G$ is the group of homeomorphisms of a topological space $X$. Let $X$ have as point set $\{t_g | g\in G\} \sqcup \{u_{g,h},v_{g,h}, w_{g,h} | (g,h) \in G\times G\}$. Define the topology to be the finest topology satisfying the following specialization conditions: for every $(g,h)\in G\times G$, impose $u_{g,h}$ specializes to $t_g$, $v_{g,h}$ specializes to $u_{g,h}$, $t_g$ and $t_h$, and $w_{g,h}$ specializes to $v_{g,h}$, $u_{g,h}$, $t_g$, $t_h$ and $t_{g\cdot h}$. | |
| Sep 4, 2015 at 18:26 | history | edited | Arturo Magidin | CC BY-SA 3.0 |
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| Sep 4, 2015 at 17:32 | history | edited | Minimus Heximus | CC BY-SA 3.0 |
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| Sep 4, 2015 at 17:06 | history | made wiki | Post Made Community Wiki by Todd Trimble | ||
| Sep 4, 2015 at 17:05 | history | edited | Ali Taghavi | CC BY-SA 3.0 |
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| Sep 4, 2015 at 16:57 | history | edited | Ali Taghavi | CC BY-SA 3.0 |
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| Sep 4, 2015 at 16:49 | history | asked | Ali Taghavi | CC BY-SA 3.0 |