If we let H be a subgroup of (X, x, T) where x is the binary operation. Show that

(i)if H is normal, then H* is normal

H* is the closure of H

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- Nov 8th 2010, 09:36 AMTurloughmackTopological Subgroup
If we let H be a subgroup of (X, x, T) where x is the binary operation. Show that

(i)if H is normal, then H* is normal

H* is the closure of H - Nov 8th 2010, 09:54 AMDrexel28
Of course this is supposed to be a topological group, right?

So, suppose that is a topological group with multiplication and inversion maps given by respectively and . It is easy to prove that (to get an idea look here where I prove the inversion part.) Now, to prove it's normal notice that the conjugation map

is continuous since since

So, using continuity we may conclude that

. Since was arbitrary it follows that