# group action on manidolds

let $G$ be a compact Lie group, and $M$ be a smooth manifold on which $G$ acts smoothly and effectively. For each $x\in M$ the isotropy group of $x$ is defined as $G_x=(g\in G:gx=x)$. Let $H$ be a closed subgroup of $G$. Due to the differential slice theorem $M_{(H)}=(x\in M:G_x$#$H$) is a smooth $G-$invariant submanifold of $M$, where $G_x$#$H$ means $G_x$ conjugate to $H$. My question is can we get $CodimM_{(H)}>1or=1$ for any nontrivial subgroup $H$ of $G$?

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You may be missing some hypotheses. For example, if the fixed locus $F$ of $G$ in $M$ (i.e., the set of points for which $G_x=G$) has codimension bigger than $1$ (which frequently happens), then setting $H=G$ will give you $F=M_{(H)}$ of higher codimension. –  Robert Bryant Apr 23 '13 at 11:43