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If we have two matrices of finite Order and we generate a group from these two matrices, then is there any tool to find order of such group. Thanks in advance

Manpreet
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  • Yes. There is nothing special about two. Given any finite set of matrices, there are algorithms to decide whether they generate a finite group and to calculate its order. But they are far too long and technical to be summarised here. You can attempt such calculations in GAP or Magma. Do you have a specific problem of this type? – Derek Holt Dec 02 '17 at 16:10

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In general, the product of elements $a$ and $b$ of order $n$ respectively $m$ can be arbitrary, even infinite, see

Examples and further results about the order of the product of two elements in a group

For information on subgroups of $GL(n,q)$ resp. $SL(n,q)$ with two generators see this MO-question, and this one.

Dietrich Burde
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  • The question was about the order of the group generated by the two matrices, not the order of the product of $a$ and $b$. – Derek Holt Dec 02 '17 at 16:07
  • Yes, I know, but with this example of two matrices $A$, $B$ of finite order, giving a product matrix of infinite order, the order of the group $\langle A,B\rangle$ is infinite. – Dietrich Burde Dec 02 '17 at 16:26
  • Yes that's true. And even if $AB$ has finite order, $\langle A,B \rangle$ might still be infinite. – Derek Holt Dec 02 '17 at 18:06