Question

For  x∈G , define Hx = {g-1Xg | g ∈G}. Under what conditions on x will  Hx ≤ G ? Further, if Hx ≤ G , will Hx _∆ G ? Give reasons for your answer.

12 Feb 2018
Answer :
Word Count : 490
We are given a group \( G \) and an element \( x \in G \). We define \( H_x \) as: \[ H_x = \{ g^{-1} x g \mid g \in G \} \] This is the *conjugacy class* of \( x \) in \( G \). Now, let's analyze the given conditions step by step. ### Condition for \( H_x \leq G \) For \( H_x \) to be a subgroup of \( G \), it must satisfy the subgroup criteria: 1. Closure: If \( a, b \in H_x \), then \( ab \in H_x \). 2. Identity Element: The identity \( e \in H_x \). 3. Inverses: If \( a
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