Let $S$ be a finite set, and $a_1$, $a_2$, and $a_3$ be three distinct involutions of $S$ such that $a_1 a_2 a_3$ is a cyclic permutation of $S$.

I am researching on whether the group generated by $a_1, a_2, a_3$ is the symmetric group of $S$, will be grateful for any suggestions on approaching the problem or related references.

Update: The above question is refuted by a counterexample given in a comment: Let $a_1=(12)(34)$, $a_2=(23)(45)$, $a_3=(34)(15)$. Then $a_1 a_2 a_3 = (13524)$, but the group generated is a subgroup of $A_5$.

Thanks to this answer, we revise the question as follows:

Let $S$ be a finite set, and $a_1$, $a_2$, and $a_3$ be three distinct odd involutions of $S$ such that $a_1 a_2 a_3$ is a cyclic permutation of $S$. Is the group generated by $a_1, a_2, a_3$ the symmetric group of $S$? (A permutation is odd if it is a composition of an odd number of transpositions.)

Update 2: The problem is fully resolved. The answers may motivate the following (interesting?) problem:

Let $S$ be a finite set. Characterize involutions $a_1$, $a_2$, and $a_3$ of $S$ that generate the symmetric group of $S$.

Let $a_1$, $a_2$, and $a_3$ be three involutions of a finite set such that $a_1 a_2 a_3$ is a cyclic permutation. Is the group generated by $a_1, a_2, a_3$ the symmetric group?

Let $S$ be a finite set, and $a_1$, $a_2$, and $a_3$ be three distinct involutions of $S$ such that $a_1 a_2 a_3$ is a cyclic permutation of $S$.

I am researching on whether the group generated by $a_1, a_2, a_3$ is the symmetric group of $S$, will be grateful for any suggestions on approaching the problem or related references.

Update: The above question is refuted by a counterexample given in a comment: Let $a_1=(12)(34)$, $a_2=(23)(45)$, $a_3=(34)(15)$. Then $a_1 a_2 a_3 = (13524)$, but the group generated is a subgroup of $A_5$.

Thanks to this answer, we revise the question as follows:

Let $S$ be a finite set, and $a_1$, $a_2$, and $a_3$ be three distinct odd involutions of $S$ such that $a_1 a_2 a_3$ is a cyclic permutation of $S$. Is the group generated by $a_1, a_2, a_3$ the symmetric group of $S$? (A permutation is odd if it is a composition of an odd number of transpositions.)

Indeed, this problem appears in my research while I am not an algebraist, so any reference suggestions or group theoretic tool are welcome, not necessarily an answer.

Let $S$ be a finite set, and $a_1$, $a_2$, and $a_3$ be three distinct involutions of $S$ such that $a_1 a_2 a_3$ is a cyclic permutation of $S$.

I am researching on whether the group generated by $a_1, a_2, a_3$ is the symmetric group of $S$, will be grateful for any suggestions on approaching the problem or related references.

Update: The above question is refuted by a counterexample given in a comment: Let $a_1=(12)(34)$, $a_2=(23)(45)$, $a_3=(34)(15)$. Then $a_1 a_2 a_3 = (13524)$, but the group generated is a subgroup of $A_5$.

Thanks to this answer, we revise the question as follows:

Let $S$ be a finite set, and $a_1$, $a_2$, and $a_3$ be three distinct odd involutions of $S$ such that $a_1 a_2 a_3$ is a cyclic permutation of $S$. Is the group generated by $a_1, a_2, a_3$ the symmetric group of $S$? (A permutation is odd if it is a composition of an odd number of transpositions.)

Update 2: The problem is fully resolved. The answers may motivate the following (interesting?) problem:

Let $S$ be a finite set. Characterize involutions $a_1$, $a_2$, and $a_3$ of $S$ that generate the symmetric group of $S$.

Let $S$ be a finite set, and $a_1$, $a_2$, and $a_3$ be three distinct involutions of $S$ such that $a_1 a_2 a_3$ is a cyclic permutation of $S$.

I am researching on whether the group generated by $a_1, a_2, a_3$ is the symmetric group of $S$, will be grateful for any suggestions on approaching the problem or related references.

Let $S$ be a finite set, and $a_1$, $a_2$, and $a_3$ be three distinct involutions of $S$ such that $a_1 a_2 a_3$ is a cyclic permutation of $S$.

I am researching on whether the group generated by $a_1, a_2, a_3$ is the symmetric group of $S$, will be grateful for any suggestions on approaching the problem or related references.

Update: The above question is refuted by a counterexample given in a comment: Let $a_1=(12)(34)$, $a_2=(23)(45)$, $a_3=(34)(15)$. Then $a_1 a_2 a_3 = (13524)$, but the group generated is a subgroup of $A_5$.

Thanks to this answer, we revise the question as follows:

Let $S$ be a finite set, and $a_1$, $a_2$, and $a_3$ be three distinct odd involutions of $S$ such that $a_1 a_2 a_3$ is a cyclic permutation of $S$. Is the group generated by $a_1, a_2, a_3$ the symmetric group of $S$? (A permutation is odd if it is a composition of an odd number of transpositions.)

Indeed, this problem appears in my research while I am not an algebraist, so any reference suggestions or group theoretic tool are welcome, not necessarily an answer.