# normal subgroup...

• Nov 23rd 2009, 02:02 AM
TTB
normal subgroup...
Let r be a subgroup of G.

Let Ar be the set of all products (finite) of elements of the form (gi ri gi^-1) or (gi ri^-1 gi^-1) i = 1 to n.

How can one show that Ar is normal in G.

Thanks x
• Nov 23rd 2009, 02:22 AM
tonio
Quote:

Originally Posted by TTB
Let r be a subgroup of G.

Let Ar be the set of all products (finite) of elements of the form (gi ri gi^-1) or (gi ri^-1 gi^-1) i = 1 to n.

How can one show that Ar is normal in G.

Thanks x

Isn't it clear this question belongs in "Abstract Algebra and Linear Algebra"??

Anyway, this is trivial when one realizes that \$\displaystyle x^{-1}(ab)x=(x^{-1}ax)(x^{-1}bx)\$. The set that you denoted Ar is more usually denoted by \$\displaystyle r^G\$ and it's called
the normal closure of the subgroup r in G.
It also way more usual to denote sets, groups, subgroups, etc. with capital letters...

Tonio
• Nov 23rd 2009, 06:41 AM
TTB
Tonio - you are right, this should be in Abstract Algebra - my apologies, i must have typed it in the wrong forum by accident. (Worried)

I know that capital letters are the norm, I just wanted to make the notation clearer.

Thanks x