This is a subgroup of S5, but it isn't generated by a 2 cycle and a 5 cycle. See here groupprops.subwiki.org/wiki/Subgroup_structure_of_symmetric_group:S5. The issue is that there a lot of subgroups of D5 that are isomorphic to Z2 and not all of them contain a 2-cycle, so knowing we have a transposition limits our options by a lot more.
Sure! I'm glad you like them. If there's any other topics you want covered feel free to suggest something, otherwise I'll be uploading some more Galois theory soon.
yo!!!! i'm getting into galois theory and these examples are paramount. great stuff, keep uploading!!!
Thanks for the support! I have a few Galois theory videos upcoming.
Another subgroup of S5 exists that contains both Z2 and Z5: the dihedral group D5 (order 10).
This is a subgroup of S5, but it isn't generated by a 2 cycle and a 5 cycle. See here groupprops.subwiki.org/wiki/Subgroup_structure_of_symmetric_group:S5.
The issue is that there a lot of subgroups of D5 that are isomorphic to Z2 and not all of them contain a 2-cycle, so knowing we have a transposition limits our options by a lot more.
@@coconutmath4928 aaah indeed, it's a double transposition that generated D5. Thank you for pointing that out.
Of course! Subgroups of S_n are tricky.
5*(2/sqrt(5))^4-4=0?
It is an example of polynomial that is not solvable by radicals
We still know the shape so not really important.
Indeed, it doesn't really affect the argument. But yes, in my head I replaced fourth root of 4/5 with square root of 4/5, which is a bit off, haha.
Enjoying your videos a lot! Thanks.
Sure! I'm glad you like them. If there's any other topics you want covered feel free to suggest something, otherwise I'll be uploading some more Galois theory soon.
I think a way to construct another polynomial not solvable by radicals from the proof.
f(0)>0, f(1)