In this post all groups are finite, and all representations are complex linear finite dimensional representations.
If a group $G$ is abelian, then all of its irreducible representations are of one dimension. The converse is also true: if all of its irreducible representations are of one dimension, then that group is abelian.
What interests me is to what extent can we detect nonabelianity from this information. To be more precise, I'd like to argue that nonabelian groups do not have $1$-dimensional representations other than the trivial one. However, this is a false statement because the permutation group $S_3$ over $3$ elements has a nontrivial yet $1$-dimensional representations.
But that representation comes from induction from its normal subgroup $A_3$! Therefore my guess should be modify so that the group is simple. The simplest nonabelian simple group is $A_5$. I also check $A_6$ and $A_7$ on Groupprop. Neither of them has nontrivial $1$-dimensional representations. So is my guess true:
Any nonabelian simple group has no nontrivial $1$-dimensional representation?
Hint: kernels of characters are normal subgroups and $G$ is simple if and only if $ker(\chi)=1$ for all non-principle irreducible characters $\chi$ of $G$. Otherwise stated, a group is simple if and only if all its non-principle characters are faithful.