Question

Let \( G \) be a group with identity element \( e \), and let \( g, h \in G \) be such that the following hold: \[ g \neq e, \quad g^2 = e, \quad h \neq e, \quad h^2 \neq e, \quad {and} \quad ghg^{-1} = h^2. \] Then, the least positive integer \( n \) for which \( h^n = e \) is (in integer).

Hint:

For elements in groups where conjugation by one element doubles the powers of another element, track the powers and find the smallest \( n \) that brings the element back to the identity.

Answer 1

Correct Answer: 3

Step 1: Analyzing the Group Properties
We are given that \( g^2 = e \), meaning \( g \) is an element of order 2 in the group \( G \). Also, \( ghg^{-1} = h^2 \), which implies that conjugation by \( g \) doubles the powers of \( h \). Step 2: Investigating Powers of \( h \)
Let's compute successive powers of \( h \): \[ ghg^{-1} = h^2, \quad g^2hg^{-2} = gghg^{-1}g^{-1} = h^4, \quad g^3hg^{-3} = gh^4g^{-1} = h^8. \] Thus, the powers of \( h \) are doubling with each conjugation by \( g \). We are looking for the least \( n \) such that \( h^n = e \). Step 3: Finding the Order of \( h \)
We find that \( h^2 = h^4 = h^8 = e \), which suggests that the order of \( h \) is 3. Step 4: Conclusion
Thus, the least positive integer \( n \) for which \( h^n = e \) is \( \boxed{3} \). \[ \boxed{3} \quad h^3 = e \]