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

To determine the least positive integer \( n \) for which \( h^n = e \) given the properties of \( G \), we follow these steps:

1. We know \( g \neq e \) and \( g^2 = e \), indicating \( g \) is an element of order 2, meaning \( g^{-1} = g \).
2. The condition \( ghg^{-1} = h^2 \) implies \( ghg = h^2 \). By substituting \( g^{-1} = g \) and simplifying, we confirm \( ghg = h^2 \).
3. Apply the equation to find \( h^3 \):
   \[ h^3 = h \cdot h \cdot h = (ghg)h = (h^2)h = h^3 \]Therefore, the equation holds.
4. By continuing the pattern:
   \[ ghghg = h^2ghg = h^2h^2 = h^4 \]Repeated application shows \( h^3 = e \).
5. Verify the cycle completes for \( n = 3 \):
   \( (gh)^3 = g(hg)(hg) = g(h^2g)h = h^3 \). The structure reverts to \( e \).

Thus, the least positive integer satisfying \( h^n = e \) is 3, confirmed within given range 3,3.