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Ali
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Let $x$ and $y$ be two element of general linear group $SL(n,q)$, such that the orders of $x$ and $y$ be some primitive prime divisor of $q^n-1$. Is it true that if $xy\not=yx$, then $x$ and $y$ generate $SL(n,q)$ ?

Note that the order of an element like $z$$x$ of $SL(n,q)$ is a primitive prime divisor of $q^n-1$ if and only if $|x|\mid (q^n-1)$ but $|x|\not\mid(q^k-1)$ for every $k<n$.

Let $x$ and $y$ be two element of general linear group $SL(n,q)$, such that the orders of $x$ and $y$ be some primitive prime divisor of $q^n-1$. Is it true that if $xy\not=yx$, then $x$ and $y$ generate $SL(n,q)$ ?

Note that the order of an element like $z$ of $SL(n,q)$ is a primitive prime divisor of $q^n-1$ if and only if $|x|\mid (q^n-1)$ but $|x|\not\mid(q^k-1)$ for every $k<n$.

Let $x$ and $y$ be two element of general linear group $SL(n,q)$, such that the orders of $x$ and $y$ be some primitive prime divisor of $q^n-1$. Is it true that if $xy\not=yx$, then $x$ and $y$ generate $SL(n,q)$ ?

Note that the order of an element like $x$ of $SL(n,q)$ is a primitive prime divisor of $q^n-1$ if and only if $|x|\mid (q^n-1)$ but $|x|\not\mid(q^k-1)$ for every $k<n$.

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Ali
Ali
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The generating graphgenerators of generalspecial linear groups

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Ali
Ali
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The generating graph of general linear groups

Let $x$ and $y$ be two element of general linear group $SL(n,q)$, such that the orders of $x$ and $y$ be some primitive prime divisor of $q^n-1$. Is it true that if $xy\not=yx$, then $x$ and $y$ generate $SL(n,q)$ ?

Note that the order of an element like $z$ of $SL(n,q)$ is a primitive prime divisor of $q^n-1$ if and only if $|x|\mid (q^n-1)$ but $|x|\not\mid(q^k-1)$ for every $k<n$.