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.

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Comments

  • 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!

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