Timeline for Groups with all normal subgroups characteristic
Current License: CC BY-SA 4.0
18 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 20, 2022 at 8:39 | comment | added | YCor | Of course it won't help in describing these groups, but the condition means that the $\mathrm{Out}(G)$-action on the lattice $\mathcal{N}(G)$ of normal subgroups of $G$ is trivial. Trivial (already mentioned) instances are when one of the two is "trivial": when $\mathrm{Out}(G)=\{1\}$ or when $G$ is simple. | |
| Sep 20, 2022 at 8:06 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
added the tag (normal-subgroups)
|
| Jul 6, 2015 at 20:50 | answer | added | Marty Isaacs | timeline score: 10 | |
| Jan 11, 2011 at 10:11 | comment | added | Someone | Normal subgroups of complete groups are characteristic. | |
| Jan 11, 2011 at 3:24 | history | edited | Alex B. | CC BY-SA 2.5 |
added 718 characters in body
|
| Jan 2, 2011 at 2:42 | history | edited | Alex B. | CC BY-SA 2.5 |
Put semi-dihedral back in, removed speculations at the end
|
| Jan 2, 2011 at 2:32 | answer | added | Beren Sanders | timeline score: 15 | |
| Jan 1, 2011 at 17:01 | history | edited | Alex B. | CC BY-SA 2.5 |
Retracted false claim about quotients of such groups
|
| Jan 1, 2011 at 17:00 | comment | added | Alex B. | @damiano Thanks, you are absolutely right. | |
| Jan 1, 2011 at 16:51 | comment | added | Alex B. | Actually, here is a counterexample to the original claim that the property is inherited by normal subgroups: $V_4$ in $S_4$. One can probably obtain many more by taking semi-direct products of groups that don't have the property by their outer automorphism groups. | |
| Jan 1, 2011 at 16:49 | comment | added | damiano | @Alex: are you sure that quotient groups inherit the property? It seems that the group $G:=S_5 \times S_7$ has abelianization that is a product of two cyclic groups of order 2. In particular, every non-trivial subgroup of the abelianization is normal, but not characteristic. On the other hand, if I am not mistaken, the only normal subgroups of $G$ are also characteristic. | |
| Jan 1, 2011 at 16:33 | comment | added | Alex B. | @BS I am sorry, that was a mistake. I meant quotient groups and not normal subgroups. I can't think of any counter examples to the latter, but nor can I see why it should be true. | |
| Jan 1, 2011 at 16:31 | history | edited | Alex B. | CC BY-SA 2.5 |
deleted 88 characters in body
|
| Jan 1, 2011 at 16:17 | comment | added | BS. | Could you explain why the property is inherited by normal subgroups ? | |
| Jan 1, 2011 at 16:07 | history | edited | Alex B. | CC BY-SA 2.5 |
Corrected wrong claim about 2-groups; edited title; added 1 characters in body
|
| Jan 1, 2011 at 15:48 | comment | added | Torsten Ekedahl | Any subgroup of index 2 in the quaternion group of order 8. | |
| Jan 1, 2011 at 15:44 | comment | added | Gerry Myerson | Sorry I can't help, but, if it's not too much trouble, can someone give an example of a normal subgroup that isn't characteristic? | |
| Jan 1, 2011 at 15:38 | history | asked | Alex B. | CC BY-SA 2.5 |