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Thread: Conjugacy in finite groups

  1. #1
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    Conjugacy in finite groups

    Let $\displaystyle A, B$ be finite groups with $\displaystyle H \subset A \cap B$ be subgroup.
    Suppose there exists a finite number of elements $\displaystyle h_{1}, h_{2}, \ldots, h_{r} \in H$ such that $\displaystyle a \sim_{A} h_{1} \sim_{B} h_{2} \sim_{A} \ldots \sim_{A(B)} h_{r}$ where $\displaystyle a \in A$.

    I know that $\displaystyle a, h_{1}, h_{2}, \ldots, h_{r}$ all have same order.
    But it doesn't mean $\displaystyle a, h_{r}$ in the same conjugacy class.

    Is there anyway to obtain $\displaystyle a \sim_{A} h_{r}$ or $\displaystyle a, h_{r}$ in the same conjugacy class without adding abelian or self-conjugate?
    Last edited by deniselim17; Sep 26th 2017 at 05:42 AM.
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  2. #2
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    Re: Conjugacy in finite groups

    The only way I can understand your hypothesis is to assume both A and B are subgroups of a group G. In this case, it is obvious that a is conjugate to $h_r$ in G since conjugacy is an equivalence relation. Assuming $a\in A\cap B$, are you asking under what conditions is a conjugate to $h_r$ in A?
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  3. #3
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    Re: Conjugacy in finite groups

    Yes, $\displaystyle G=A^{*}_{H} B$. $\displaystyle G$ is the free product of groups $\displaystyle A$ and $\displaystyle B$ amalgamating subgroup $\displaystyle H$.
    I have $\displaystyle A, B$ are finite.
    I have results where $\displaystyle {x}^{A} \cap <y> =\emptyset, {x}^{B} \cap <y> =\emptyset$, where $\displaystyle ||x||=0, ||y|| \leq 1$.

    TO SHOW $\displaystyle {x}^{G} \cap <y>=\emptyset$, I prove it by contradiction.

    Suppose $\displaystyle {x}^{G} \cap <y> \neq \emptyset$, so I have $\displaystyle x \sim_{G} y^{k}$, where $\displaystyle k \in \mathbb{Z}$.
    By Lemma, there exists a finite number of elements $\displaystyle h_{i} \in H$ such that $\displaystyle y^{k} \sim_{A} h_{1} \sim_{B} h_{2} \sim_{A} \ldots \sim_{A(B)} h_{r}=x$.

    To show it by contradiction, I need to have $\displaystyle y^{k} \sim_{A} x$ or $\displaystyle y^{k} \sim_{B} x$. I just can't see how to reach this step.
    Last edited by deniselim17; Sep 26th 2017 at 06:15 PM.
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