$2^n$ denotes elementary abelian 2-groups of rank n.
I am reading a paper which has $2^4.2^3 = 2^5.2^2$
Are these two extensions equal? Are there any facts I am not aware of? Because I checked an example: $Z_p$ extended by $Z_p$ x $Z_p$ can be nonabelian. So what if the left is elementary abelian and the right is not abelian.
$2^4.2^3$ does not denote a single group, it denotes an arbitrary group $G$ with an elementary abelian normal subgroup $N$ of order $2^4$, such that $G/N$ is elementary abelian of order $2^3$.
Similarly for $2^5.2^2$, and the set of isomorphism classes of groups with structure $2^4.2^3$ is certainly not equal to the set with structure $2^5.2^2$, although of course some groups, such as an elementary abelian group of order $2^7$ lie in both sets.
I suspect that in the paper that you are reading there is more information about the specific group in question, which might well fall into both classes.
