## The diagonal subgroup is not normal in the direct square of Sym(3)

Show that the diagonal subgroup $\{ (\sigma,\sigma) \ |\ \sigma \in S_3 \}$ is not normal in $S_3 \times S_3$.

Consider $(1, (1\ 2)) \in S_3 \times S_3$ and $((1\ 3),(1\ 3)) \in D$. We have $(1, (1\ 2))((1\ 3), (1\ 3))(1, (1\ 2)) = ((1\ 3), (2\ 3)) \notin D$, so that $D$ is not normal in $S_3 \times S_3$.

• rip  On February 18, 2011 at 3:30 pm

Perhaps you want to change the title: the diagonal subgroup is _not_ normal….

• nbloomf  On February 18, 2011 at 11:03 pm

Thanks!

• Gobi Ree  On November 22, 2011 at 1:41 am

There seems an error: $(1\ 2)(1\ 3)(1\ 2) = (2\ 3)$.

• nbloomf  On November 22, 2011 at 9:51 am

Fixed. Thanks!