Help on the following 4 true/false question on absract algbra.
1) The rotation of a Diheral group form a cyclic subgroup.
2)A subgroup of a cyclic group is a cyclic
3)A subset H of a group is a subgroup if a and b belong to H implies that ab^-1 belong to H.
4) If A is a 21 cycle in S50, the order of A is 21.

I personally thinking is all true.

However I am not confident, can someone double check for me.

Question

Help on the following 4 true/false question on absract algbra.
1) The rotation of a Diheral group form a cyclic subgroup.
2)A subgroup of a cyclic group is a cyclic
3)A subset H of a group is a subgroup if a and b belong to H implies that ab^-1 belong to H.
4) If A is a 21 cycle in S50, the order of A is 21.

I personally thinking is all true.

However I am not confident, can someone double check for me.

anonymous · Answer

rotations are cyclic i am fairly sure
generated by the first rotation

anonymous · Answer

subgroup of a cyclic group, hmmm cyclic group iso to $\mathbb{Z}_n$ if i am not mistaken so that would seem to be cyclic too generated by element of least order

anonymous · Answer

3 is definitely true

anonymous · Answer

last one... you are on your own sorry, maybe someone else could chime in

anonymous · Answer

Order of a cycle equals its length so I guess the last one is true as well.