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

Step 1: Use the given conjugation relation.
We are given:

• \( g^2 = e \), so \( g^{-1} = g \).
• \( ghg^{-1} = h^2 \).

Since \( g^{-1} = g \), the conjugation relation can be rewritten as:

\[ ghg = h^2. \]

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Step 2: Apply conjugation repeatedly using the fact \( g^2 = e \).
Conjugate both sides again by \( g \):

\[ g(ghg)g = g(h^2)g. \]

Using \( g^2 = e \), the left-hand side simplifies to:

\[ h = gh^2g. \]

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Step 3: Express the right-hand side using the conjugation rule.
From the given rule,

\[ gh^2g = (ghg)^2 = (h^2)^2 = h^4. \]

Thus we obtain:

\[ h = h^4. \]

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Step 4: Deduce the order of \( h \).
From \( h = h^4 \), we get:

\[ h^3 = e. \]

Hence, the order of \( h \) divides 3. Since \( h \neq e \), the smallest positive integer \( n \) such that \( h^n = e \) is:

\[ \boxed{3}. \]

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Final Answer: \(\boxed{3}\)