# Difference between revisions of "Orbit"

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An '''orbit''' is part of a [[set]] on which a [[group]] [[group action|acts]]. | An '''orbit''' is part of a [[set]] on which a [[group]] [[group action|acts]]. | ||

− | Let <math>G</math> be a group, and let <math>S</math> be a <math>G</math>-set. The '''orbit''' of an element <math>x\in S</math> is the set <math>Gx</math>, i.e., the set of [[conjugate (group theory) | conjugate]]s of <math>x</math>, or the set of elements <math>y</math> in <math>S</math> for which there exists <math>\alpha \in G</math> for which <math>\alpha x = y</math>. | + | Let <math>G</math> be a group, and let <math>S</math> be a <math>G</math>-set. The '''orbit''' of an element <math>x\in S</math> is the set <math>Gx</math>, i.e., the set of [[conjugate (group theory) | conjugate]]s of <math>x</math>, or the set of elements <math>y</math> in <math>S</math> for which there exists <math>\alpha \in G</math> for which <math>\alpha\cdot x = y</math>. |

− | For <math>x\in S</math>, the mapping <math>\alpha \mapsto \alpha x</math> is sometimes known as the ''orbital mapping defined by <math>x</math>''; it is a homomorphism of the <math>G</math>-set <math>G</math> (with action on itself, by left translation) into <math>S</math>; the image of <math>G</math> is the orbit of <math>x</math>. We say that <math>G</math> acts ''freely'' on <math>S</math> if the orbital mapping defined by <math>x</math> is [[injective]], for all <math>x \in S</math>. | + | For <math>x\in S</math>, the mapping <math>\alpha \mapsto \alpha\cdot x</math> is sometimes known as the ''orbital mapping defined by <math>x</math>''; it is a homomorphism of the <math>G</math>-set <math>G</math> (with action on itself, by left translation) into <math>S</math>; the image of <math>G</math> is the orbit of <math>x</math>. We say that <math>G</math> acts ''freely'' on <math>S</math> if the orbital mapping defined by <math>x</math> is [[injective]], for all <math>x \in S</math>. |

The set of orbits of <math>S</math> is the [[quotient set]] of <math>S</math> under the relation of conjugation. This set is denoted <math>G\backslash S</math>, or <math>S/G</math>. (Sometimes the first notation is used when <math>G</math> acts on the left, and the second, when <math>G</math> acts on the right.) | The set of orbits of <math>S</math> is the [[quotient set]] of <math>S</math> under the relation of conjugation. This set is denoted <math>G\backslash S</math>, or <math>S/G</math>. (Sometimes the first notation is used when <math>G</math> acts on the left, and the second, when <math>G</math> acts on the right.) | ||

− | Let <math>G</math> be a set acting on <math>S</math> from the right, and let <math>H</math> be a [[normal subgroup]] of <math>G</math>. Then <math>G</math> acts on <math>S/H</math> from the right, under the action <math> | + | Let <math>G</math> be a set acting on <math>S</math> from the right, and let <math>H</math> be a [[normal subgroup]] of <math>G</math>. Then <math>G</math> acts on <math>S/H</math> from the right, under the action <math>xH\cdot g = xHg = xgH</math>, for <math>x\in S</math>. (<math>H</math> acts trivially on this set, so <math>(E/H)/G = (E/H)/(G/H)</math>.) Consider the canonical mapping <math>\phi : E/H \to E/G</math>. The inverse images of elements of <math>E/G</math> under <math>\phi</math> are the orbits of <math>E/H</math> under action of <math>G</math>; thus on passing to the quotient, <math>\phi</math> defines an isomorphism from <math>(E/H)/G</math> to <math>E/G</math>. |

Suppose <math>G</math> and <math>H</math> are groups, and <math>G</math> acts on <math>S</math> on the left, and <math>H</math> on the right; suppose furthermore that the operations of <math>G</math> and <math>H</math> commute, i.e., for all <math>g\in G</math>, <math>h\in H</math>, <math>x\in S</math>, | Suppose <math>G</math> and <math>H</math> are groups, and <math>G</math> acts on <math>S</math> on the left, and <math>H</math> on the right; suppose furthermore that the operations of <math>G</math> and <math>H</math> commute, i.e., for all <math>g\in G</math>, <math>h\in H</math>, <math>x\in S</math>, | ||

− | <cmath> ( | + | <cmath> (g\cdot x)\cdot h = g\cdot (x\cdot h) .</cmath> |

Let <math>H^0</math> be the opposite group of <math>G</math>; then the actions of <math>G</math> and <math>H</math> on <math>S</math> define a left action of <math>G \times H^0</math> on <math>S</math>. The set <math>(G\times H^0)\backslash S</math> is denoted <math>G \backslash S /H</math>. Since <math>G</math> and <math>H^0</math> are normal subgroups of <math>G \times H^0</math>, by the previous paragraph, the <math>G</math>-sets <math>(G \backslash S)/H</math>, <math>G \backslash ( S/H)</math>, <math>G \backslash S /H</math> are isomorphic and identitfied with each other. | Let <math>H^0</math> be the opposite group of <math>G</math>; then the actions of <math>G</math> and <math>H</math> on <math>S</math> define a left action of <math>G \times H^0</math> on <math>S</math>. The set <math>(G\times H^0)\backslash S</math> is denoted <math>G \backslash S /H</math>. Since <math>G</math> and <math>H^0</math> are normal subgroups of <math>G \times H^0</math>, by the previous paragraph, the <math>G</math>-sets <math>(G \backslash S)/H</math>, <math>G \backslash ( S/H)</math>, <math>G \backslash S /H</math> are isomorphic and identitfied with each other. | ||

## Latest revision as of 15:44, 7 September 2008

An **orbit** is part of a set on which a group acts.

Let be a group, and let be a -set. The **orbit** of an element is the set , i.e., the set of conjugates of , or the set of elements in for which there exists for which .

For , the mapping is sometimes known as the *orbital mapping defined by *; it is a homomorphism of the -set (with action on itself, by left translation) into ; the image of is the orbit of . We say that acts *freely* on if the orbital mapping defined by is injective, for all .

The set of orbits of is the quotient set of under the relation of conjugation. This set is denoted , or . (Sometimes the first notation is used when acts on the left, and the second, when acts on the right.)

Let be a set acting on from the right, and let be a normal subgroup of . Then acts on from the right, under the action , for . ( acts trivially on this set, so .) Consider the canonical mapping . The inverse images of elements of under are the orbits of under action of ; thus on passing to the quotient, defines an isomorphism from to .

Suppose and are groups, and acts on on the left, and on the right; suppose furthermore that the operations of and commute, i.e., for all , , , Let be the opposite group of ; then the actions of and on define a left action of on . The set is denoted . Since and are normal subgroups of , by the previous paragraph, the -sets , , are isomorphic and identitfied with each other.

Let be a group, and a subgroup of ; let it act on from the right. Then the set is the set of left cosets mod .

If is a group and are subgroups of , then the set is called the set of *double cosets* mod and .