and . Show that . Deduce that if , then
If , then if .
How can I get to the point where they are equal with this information?
well it's obvious that is a subgroup of . so what we need to show is that the normalizer is a subgroup of the centralizer if |H| = 2.
so let g be in . clearly , and we know that , where h is the non-identity element of H.
but if , then gh = g, and thus h = e, contradiction. hence , so