isomorphism between direct product of general linear groups

Question

We know that for m=p_1p_2...p_n which p_i are prime numbers, then SL(2,Z_m) is isomorphic to direct productt of SL(2,p_i). Can we say that GL(2,2) is isomorph to direct product of itself ?

Answer 1

Every group $G$ is the direct product of the family of groups $\{G\}$, much like every number $n$ is the product of the numbers in the set $\{n\}$.

However, if $\operatorname{GL}(2,2)$ is a direct product of two or more groups, then all but one of those groups has size $1$, and the remaining one is isomorphic to $\operatorname{GL}(2,2)$. This is similar to the fact that if the number 2 is a product of two or more positive integers, then all but one of those positive integers is $1$ and the remaining one is $2$.

In particular, if you are asking if $\operatorname{GL}(2,2) \stackrel{?}{\cong} \operatorname{GL}(2,2) \times \operatorname{GL}(2,2)$, then the answer is no.