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Stefan Kohl ♦
Stefan Kohl ♦
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On Finite groups which have elements $g$ of order $pq$ such that the sizes of the conjugacy classclasses of finite group?$g^p$, of $g^q$ and of $g^{q-p}$ coincide

Let $G$ be a finite group and let $g\in G$ such that$g \in G$ be an element of order $o(g)=pq$$pq$, where $p<q$$p < q$ are prime prime numbers. (Also we know thatDenote by $g^G$ isthe conjugacy class of $g$ in $G$.) Under which conditions we can say that $|(g^p)^G|=|(g^q)^G|=|(g^{q−p})^G|$ does the following hold?: $$ |(g^p)^G| = |(g^q)^G| = |(g^{q−p})^G| $$ -- Is it possible that it happenthis happens?

On conjugacy class of finite group?

Let $G$ be a finite group and $g\in G$ such that $o(g)=pq$, where $p<q$ are prime numbers. (Also we know that $g^G$ is conjugacy class of $g$ in $G$.) Under which conditions we can say that $|(g^p)^G|=|(g^q)^G|=|(g^{q−p})^G|$? Is it possible that it happen?

Finite groups which have elements $g$ of order $pq$ such that the sizes of the conjugacy classes of $g^p$, of $g^q$ and of $g^{q-p}$ coincide

Let $G$ be a finite group and let $g \in G$ be an element of order $pq$, where $p < q$ are prime numbers. Denote by $g^G$ the conjugacy class of $g$ in $G$. Under which conditions does the following hold?: $$ |(g^p)^G| = |(g^q)^G| = |(g^{q−p})^G| $$ -- Is it possible that this happens?

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Anna
Anna
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On conjugacy class of finite group?

Let $G$ be a finite group and $g\in G$ such that $o(g)=pq$, where $p<q$ are prime numbers. (Also we know that $g^G$ is conjugacy class of $g$ in $G$.) Under which conditions we can say that $|(g^p)^G|=|(g^q)^G|=|(g^{q−p})^G|$? Is it possible that it happen?