I am reading up on Fraleigh's A First Course in Abstract Algebra, and he says ($H$ subgroup of $G$) $Hg=gH$ $iff$ $i_g[H]=H$ $iff$ $H$ is invariant under all inner automorphisms. I look up invariant and I find this definition:
"Firstly, if one has a group G acting on a mathematical object (or set of objects) X, then one may ask which points x are unchanged, "invariant" under the group action, or under an element g of the group." from Invariant Description Wiki.
First I am wondering if that means the elements of $H$ do not change but change positions (hence the permutation) or is $H$ the identity under all inner automorphisms of $G$. W
EDIT: Too many questions asked by me, I will ask them separately.