This question was asked in a masters exam for which I am preparing and I was unable to solve this particular question.
In the group $G$ of all invertible $4\times 4$ matrices with entries in the field of $3$ elements, any $3$-Sylow subgroup has what cardinality?
Cardinality of group $G$ is $80 \times 78 \times 72 \times 54$. I thought $3$-Sylow subgroup has cardinality $3^n =81$ but this is wrong .
Answer:
$$729$$
I am not able to think any other result which I can use it here. So, Can anyone please tell me right way to approach this question?
Thanks!!