Timeline for $2$-group with two isomorphic normal subgroups of index $4$ with non-isomorphic quotients
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| Oct 19, 2014 at 14:00 | history | edited | Ricardo Andrade |
removed tag 'normal-groups'
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| S Apr 25, 2014 at 3:22 | history | bounty ended | CommunityBot | ||
| S Apr 25, 2014 at 3:22 | history | notice removed | CommunityBot | ||
| Apr 17, 2014 at 17:10 | comment | added | Yassine Guerboussa | @Jeremy Rickard: you are right, thank you. | |
| Apr 17, 2014 at 12:46 | comment | added | Jeremy Rickard | @YassineGuerboussa: If the isomorphism between maximal subgroups of $N$ and $M$ is induced by an automorphism of $G$ then the quotients will be isomorphic. Of course, this doesn't rule out the possibility that there's an isomorphism not induced by an automorphism of $G$, but according to my amateurish Magma calculations this isn't the case for the free Burnside group of exponent four on two generators. | |
| Apr 17, 2014 at 10:12 | comment | added | Yassine Guerboussa | Ok, I think in a finite $p$-group $G$ such that $Aut(G)$ induces the full linear group on $G/\Phi(G)$, all the maximal subgroups of $G$ are isomorphic. Any relativley free finite $p$-group have the property above. I don't know if one can choose two maximal subgroups $N$ and $M$ such that the isomorphism between $N$ and $M$ (induced by an automorphism of $G$) takes a $G$ invariant maximal subgroup of $N$ whose quotient has exponent $4$ to a subgroup of $M$ whose quotient has exponent $2$. Another thing, any free Burnside group of exponent 4 is finite (thanks to Sanov). | |
| Apr 17, 2014 at 2:24 | comment | added | Russ Woodroofe | This question has a somewhat similar flavor to mathoverflow.net/questions/153433/… | |
| S Apr 17, 2014 at 1:56 | history | bounty started | verret | ||
| S Apr 17, 2014 at 1:56 | history | notice added | verret | Draw attention | |
| Apr 17, 2014 at 1:55 | comment | added | verret | I'm not sure I understand your example. Could you maybe be more explicit? (Keep in mind that I am only considering finite groups.) | |
| Apr 16, 2014 at 19:06 | comment | added | Yassine Guerboussa | I wonder if a 2-generated 2-group with all maximal subgroups isomorphic, and a non elementary abelian abelianized do the claim. Did you checked the free Burnside group on 2-generators and exponent 4. | |
| Apr 14, 2014 at 6:17 | history | asked | verret | CC BY-SA 3.0 |