Stabilizer

of an element $a$ in a set $M$

The subgroup $G_a$ of a group of transformations $G$, operating on a set $M$, (cf. Group action) consisting of the transformations that leave the element $a$ fixed: $G_a = \{ g \in G : ag = a \}$. The stabilizer of $a$ is also called the isotropy group of $a$, the isotropy subgroup of $a$ or the stationary subgroup of $a$. If $b \in M$ is in the orbit of $a$, so $b = af$ with $f \in G$, then $G_b = f^{-1}G_af$. If one considers the action of the group $G$ on itself by conjugation, the stabilizer of the element $g \in G$ will be the centralizer of this element in $G$; if the group acts by conjugation on the set of its subgroups, then the stabilizer of a subgroup $H$ will be the normalizer of this subgroup (cf. Normalizer of a subset).

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In case $M$ is a set of mathematical structures, for instance a set of lattices in $\mathbf{R}^n$, on which a group $G$ acts, for instance the group of Euclidean motions, then the isotropy subgroup $G_m$ of $m \in M$ is the symmetry group of the structure $m \in M$.

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