Perhaps something I am studying about the "Coset Enumeration", though probably "cyclic conjugate" in group theory seems to be a much general term.

In particular I want to find the "cyclic conjugate" of a set of union of relators and their inverses of a finitely presented group.

Seems like nobody in the room right now. Ping me if anyone can help me with the definition of this.

@gxyd Is this really about elements of a set and not about words (finite strings formed from elements of a given sets)?

Yes, this is about "words" precisely, to repeat I meant: the elements of set are "words" in generators and their inverses.

When I tried Google and Google Books, most results seem to be about words.

May be the definition for a set of "words" is something similar to that of definition for individual "words".

"Given a word $w=a_1a_2\cdots a_n$ on a set $\Sigma$ (may or may not be finite), a cyclic conjugate of $w$ is a word $v$ derived from $w$ based on a cyclic permutation. In other words, $v=\pi(a_1)\pi(a_2)\cdots \pi(a_n)$ for some cyclic permutation $\pi$ on $\lbrace a_1,\ldots, a_n\rbrace$. Equivalently, $v$ and $w$ are cyclic conjugates of one another iff $w=st$ and $v=ts$ for some words $s,t$."

This is written on PlanetMath: planetmath.org/cyclicpermutation

So if I understand it correctly, the cyclic conjugates of $abca$ would be $abca$, $bcaa$, $caab$ and $aabc$.

Similar example is given on PlanetMath: For example, the cyclic conjugates of the word $ababa$ over $\lbrace a,b\rbrace$ are $$baba^2,\quad aba^2b,\quad ba^2ba,\quad a^2bab,\quad\mbox{and itself}.$$

It seems quite intuitive.

So for the set would be the union of cyclic conjugates of all elements.

Thank you very @MartinSleziak especially for the link.

Not much to thank for - I simply tried whether I can find something relevant using google.

To be true, I didn't thought it to find in something for permutations. But after you sent me the link, I thought the term "cyclic" should have made me to look for it, even in the permutations related search results.

hi there, can someone help me oh this stuff? math.stackexchange.com/questions/851669/…