Notation question: $SL(2,3)$, $SL_2(F_3)$, and $SL_2(\mathbb{Z}/3\mathbb{Z})$?

Question

Do this all notation equivalent? $SL(2,3)$, $SL_2(F_3)$, and $SL_2(\mathbb{Z}/3\mathbb{Z})$

I am new to the topic and want to know whether above expressions are same.

Answer 1

Yes, all notations are the same.

The general form of the second and third variant is $\operatorname{SL}_n(R)$, denoting the group of all $n\times n$ matrices over $R$ of determinant $1$ ("special linear group"). Since the two rings $F_3$ and $\mathbb{Z}/3\mathbb{Z}$ are isomorphic (field of order $3$), both variants denote the same thing.

The general form of the first version is $\operatorname{SL}(n,q)$, which serves as an abbreviation for $\operatorname{SL}_n(\mathbb{F}_q)$.