Let

be a group and suppose

is a cyclic subgroup which is

*not* normal. I am unsure how this question generalises, but I am specifically interested in the cyclic case so it is posed thataway.

Now, let

be a set of (right) coset representatives of

. My question is this,

-Does there always exist some such set

where every

does not contain any copies of

?

My first thought was...well, obvously there must! That is the point of the cosets! However, when I thought about it my opinion changed. For example,

, the free group generated by

and

. Let

. Then what would be the representative for, say,

? Or even

? Is

essentially just all the words which do not end in an

up to free equivalence?

And so I was wondering if anyone could clear this up for me.

Also, another reason the answer should be `No' is Burnside's Problem (as then one can just `pull' every generator out of a product and induct to get a positive solution, which is a contradiction).