I am currently reading http://www.math.uconn.edu/~kconrad/blurbs/grouptheory/group12.pdf, and have a quick question about the group being non-abelian. Let me explain:
Let $|G|=12=2^2\cdot 3$ and let $n_2,$ $n_3$ denote the number of Sylow $2,3$-subgroups respectively. By using Sylow's Theorems and counting elements we see that either $n_3=1$, or $n_3=4$ (and $n_2=1$). So we always have a normal Sylow $2$-subgroup or a normal Sylow $3$-subgroup.
On page 2 the author turns to the case $n_2\ne 1$ (so $n_2=3$, $n_3=1$) and says "Since $n_2\ne 1$, the group is non-abelian". Why is this true?