Group theory 32: Subgroups of free groups

[@swaruppaul4417]

Thank you so much professor for these lectures! It’s my earnest request, please give few lectures on Galois Theory.

[@michakucharczyk9460]

awesome set of lectures! is there gonna be any more of them, as suggested?

[@rudolfquine6448]

Why is index n subgroup of G is same as transitive action of G on n points, one marked?

[@DanielMT1997]

The elements fixing a given point make a subgroup of G. The action is transitive, so there is only one orbit and you can reach all the points. Now, two elements lie on the same coset if and only if they move the fixed point in the same way (they take it to the same point). As there are n options for moving the point, we have n cosets, which is the index of the subgroup.

[@l.JAI.SHREE.RAM.l]

Sir, I have a doubt, at many places group is represented by G how can it be possible, G may be a set not group, group should always denoted by (G,*). Please clear it🙏

[@expl0s10n]

If no operation is specified for a group G, we should understand it as the most natural operation that makes G a group. For example, the group Z refers to (Z,+), while the group Z_n^* should be equipped with multiplication modulo n.