- #1

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## Homework Statement

Show that if ##H## is a subgroup of ##G##, then ##H \le N_G (H)##

## Homework Equations

## The Attempt at a Solution

Essentially, we need to show that ##H \subseteq N_G (H)##; since they are both groups under the same binary operation the fact that they are subgroups will result. So let ##h \in H##. We want to show that ##h \in N_G (H)##, i.e. we want to show that ##hHh^{-1} = H##. But since the map ##\sigma_h (x) = hxh^{-1}## is automorphism an and thus a permutation from ##H## to ##H##, the set ##hHh^{-1}## is just a permuted version of ##H##, and so by set theory ##hHh^{-1} = H##, and ##h \in N_G (H)##.