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Primitive permutation group

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In <a href="Mathematics" title="Mathematics">mathematics</a>, a <a href="Permutation_group" title="Permutation group">permutation group</a> G <a href="Group_action" title="Group action">acting</a> on a non-empty finite set X is called primitive if G acts <a href="Transitive_action" class="mw-redirect" title="Transitive action">transitively</a> on X and the only <a href="Partition_of_a_set" title="Partition of a set">partitions</a> the G-action preserves are the trivial partitions into either a single set or into |X| singleton sets. Otherwise, if G is transitive and G does preserve a nontrivial partition, G is called imprimitive.

While primitive permutation groups are transitive, not all transitive permutation groups are primitive. The simplest example is the <a href="Klein_four-group" title="Klein four-group">Klein four-group</a> acting on the vertices of a square, which preserves the partition into diagonals. On the other hand, if a permutation group preserves only trivial partitions, it is transitive, except in the case of the <a href="Trivial_group" title="Trivial group">trivial group</a> acting on a 2-element set. This is because for a non-transitive action, either the <a href="Orbit_(group_theory)" class="mw-redirect" title="Orbit (group theory)">orbits</a> of G form a nontrivial partition preserved by G, or the group action is trivial, in which case all nontrivial partitions of X (which exists for |X| ≥ 3) are preserved by G.

This terminology was introduced by <a href="%C3%89variste_Galois" title="Évariste Galois">Évariste Galois</a> in his last letter, in which he used the French term équation primitive for an equation whose <a href="Galois_group" title="Galois group">Galois group</a> is primitive.<a href="#cite_note-1">[1]</a>

<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>

In the same letter in which he introduced the term "primitive", Galois stated the following theorem:<a href="#cite_note-2">[2]</a><blockquote>If G is a primitive <a href="Solvable_group" title="Solvable group">solvable group</a> acting on a finite set X, then the order of X is a power of a <a href="Prime_number" title="Prime number">prime number</a> p. Further, X may be identified with an <a href="Affine_space" title="Affine space">affine space</a> over the <a href="Finite_field" title="Finite field">finite field</a> with p elements, and G acts on X as a subgroup of the <a href="Affine_group" title="Affine group">affine group</a>.</blockquote>If the set X on which G acts is finite, its cardinality is called the degree of G.

A corollary of this result of Galois is that, if p is an odd prime number, then the order of a solvable transitive group of degree p is a divisor of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(p-1).}">

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<annotation encoding="application/x-tex">{\displaystyle p(p-1).}</annotation>

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</math><img src="./bc7d8799913b9dd4aeb920e71ef64b35cc1b600e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:8.887ex; height:2.843ex;" alt="{\displaystyle p(p-1).}" loading="lazy"> In fact, every transitive group of prime degree is primitive (since the number of elements of a partition fixed by G must be a divisor of p), and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(p-1)}">

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<annotation encoding="application/x-tex">{\displaystyle p(p-1)}</annotation>

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</math><img src="./cff815284a731edb3a54121e8c45b6bd66f8c650.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:8.24ex; height:2.843ex;" alt="{\displaystyle p(p-1)}" loading="lazy"> is the cardinality of the affine group of an affine space with p elements.

It follows that, if p is a prime number greater than 3, the <a href="Symmetric_group" title="Symmetric group">symmetric group</a> and the <a href="Alternating_group" title="Alternating group">alternating group</a> of degree p are not solvable, since their order are greater than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(p-1).}">

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<annotation encoding="application/x-tex">{\displaystyle p(p-1).}</annotation>

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</math><img src="./bc7d8799913b9dd4aeb920e71ef64b35cc1b600e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:8.887ex; height:2.843ex;" alt="{\displaystyle p(p-1).}" loading="lazy"> <a href="Abel%E2%80%93Ruffini_theorem" title="Abel–Ruffini theorem">Abel–Ruffini theorem</a> results from this and the fact that there are polynomials with a symmetric Galois group.

An equivalent definition of primitivity relies on the fact that every transitive action of a group G is isomorphic to an action arising from the canonical action of G on the set G/H of <a href="Coset" title="Coset">cosets</a> for H a subgroup of G. A group action is primitive if it is isomorphic to G/H for a <a href="Maximal_subgroup" title="Maximal subgroup">maximal subgroup</a> H of G, and imprimitive otherwise (that is, if there is a proper subgroup K of G of which H is a proper subgroup). These imprimitive actions are examples of <a href="Induced_representation" title="Induced representation">induced representations</a>.

The numbers of primitive groups of small degree were stated by <a href="Robert_Carmichael" class="mw-redirect" title="Robert Carmichael">Robert Carmichael</a> in 1937:

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There are a large number of primitive groups of degree 16. As Carmichael notes, all of these groups, except for the <a href="Symmetric_group" title="Symmetric group">symmetric</a> and <a href="Alternating_group" title="Alternating group">alternating</a> group, are subgroups of the <a href="Affine_group" title="Affine group">affine group</a> on the 4-dimensional space over the 2-element <a href="Finite_field" title="Finite field">finite field</a>.

<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

<ul><li>Consider the <a href="Symmetric_group" title="Symmetric group">symmetric group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{3}}">

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<annotation encoding="application/x-tex">{\displaystyle S_{3}}</annotation>

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</math><img src="./70e15f3e200aaa247f69c43110cc5a09ecc91b89.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{3}}" loading="lazy"> acting on the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=\{1,2,3\}}">

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<annotation encoding="application/x-tex">{\displaystyle X=\{1,2,3\}}</annotation>

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</math><img src="./955caac5ab925f254a1926536391241bb76f56e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.959ex; height:2.843ex;" alt="{\displaystyle X=\{1,2,3\}}" loading="lazy"> and the permutation</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle \eta ={\begin{pmatrix}1&2&3\\2&3&1\end{pmatrix}}.}</annotation>

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</math><img src="./f7963ee7208a0aa79180829605eb7d6203345f7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.22ex; height:6.176ex;" alt="{\displaystyle \eta ={\begin{pmatrix}1&2&3\\2&3&1\end{pmatrix}}.}" loading="lazy"></dd></dl>

Both <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{3}}">

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<annotation encoding="application/x-tex">{\displaystyle S_{3}}</annotation>

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</math><img src="./70e15f3e200aaa247f69c43110cc5a09ecc91b89.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{3}}" loading="lazy"> and the group generated by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta }">

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<annotation encoding="application/x-tex">{\displaystyle \eta }</annotation>

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</math><img src="./e4d701857cf5fbec133eebaf94deadf722537f64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }" loading="lazy"> are primitive.

<ul><li>Now consider the <a href="Symmetric_group" title="Symmetric group">symmetric group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{4}}">

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<annotation encoding="application/x-tex">{\displaystyle S_{4}}</annotation>

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</math><img src="./fc1632bc1d95d33ccb5473b9d8cc333c2dd0d13a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{4}}" loading="lazy"> acting on the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,2,3,4\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{1,2,3,4\}}</annotation>

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</math><img src="./ebf4ca66fd59843b349aed8ffa7655c1aae77625.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.077ex; height:2.843ex;" alt="{\displaystyle \{1,2,3,4\}}" loading="lazy"> and the permutation</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle \sigma ={\begin{pmatrix}1&2&3&4\\2&3&4&1\end{pmatrix}}.}</annotation>

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</math><img src="./a545243dc81a7959b66c86c8638d79ff31605b0a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.865ex; height:6.176ex;" alt="{\displaystyle \sigma ={\begin{pmatrix}1&2&3&4\\2&3&4&1\end{pmatrix}}.}" loading="lazy"></dd></dl>

The group generated by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

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<annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation>

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</math><img src="./59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy"> is not primitive, since the partition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X_{1},X_{2})}">

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<annotation encoding="application/x-tex">{\displaystyle (X_{1},X_{2})}</annotation>

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</math><img src="./60ba8ba324660a77a8647ecd53c689536879fe58.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.8ex; height:2.843ex;" alt="{\displaystyle (X_{1},X_{2})}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}=\{1,3\}}">

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<annotation encoding="application/x-tex">{\displaystyle X_{1}=\{1,3\}}</annotation>

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</math><img src="./e6710432ccca02acd5d7cf5f2aa39c3b7c899cf4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.761ex; height:2.843ex;" alt="{\displaystyle X_{1}=\{1,3\}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{2}=\{2,4\}}">

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<annotation encoding="application/x-tex">{\displaystyle X_{2}=\{2,4\}}</annotation>

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</math><img src="./88d82e7dc75ccd97ec06f77be659437454b87a8d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.761ex; height:2.843ex;" alt="{\displaystyle X_{2}=\{2,4\}}" loading="lazy"> is preserved under <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

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<annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation>

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</math><img src="./59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy">, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma (X_{1})=X_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle \sigma (X_{1})=X_{2}}</annotation>

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</math><img src="./60bd54d86bae4d983f6f928cad930073fa023793.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.194ex; height:2.843ex;" alt="{\displaystyle \sigma (X_{1})=X_{2}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma (X_{2})=X_{1}}">

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<annotation encoding="application/x-tex">{\displaystyle \sigma (X_{2})=X_{1}}</annotation>

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</math><img src="./a242c8722fcc4ff1ddef07f8053b53f5dd9c83b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.194ex; height:2.843ex;" alt="{\displaystyle \sigma (X_{2})=X_{1}}" loading="lazy">.

<ul><li>Every transitive group of prime degree is primitive</li>

<li>The <a href="Symmetric_group" title="Symmetric group">symmetric group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{n}}">

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</math><img src="./9f049ac28d4ac8097b625f9d71c1f22b2ebd1bc4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.643ex; height:2.509ex;" alt="{\displaystyle S_{n}}" loading="lazy"> acting on the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,\ldots ,n\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{1,\ldots ,n\}}</annotation>

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</math><img src="./0401c38cf1a2e51b30b38f4b93b5285aa77f8fad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.06ex; height:2.843ex;" alt="{\displaystyle \{1,\ldots ,n\}}" loading="lazy"> is primitive for every n and the <a href="Alternating_group" title="Alternating group">alternating group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{n}}">

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</math><img src="./730f6906700685b6d52f3958b1c2ae659d2d97d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.962ex; height:2.509ex;" alt="{\displaystyle A_{n}}" loading="lazy"> acting on the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,\ldots ,n\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{1,\ldots ,n\}}</annotation>

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</math><img src="./0401c38cf1a2e51b30b38f4b93b5285aa77f8fad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.06ex; height:2.843ex;" alt="{\displaystyle \{1,\ldots ,n\}}" loading="lazy"> is primitive for every n > 2.</li></ul>

<ul><li><a href="Block_(permutation_group_theory)" title="Block (permutation group theory)">Block (permutation group theory)</a></li>

<li><a href="Jordan's_theorem_(symmetric_group)" title="Jordan's theorem (symmetric group)">Jordan's theorem (symmetric group)</a></li>

<li><a href="O'Nan%E2%80%93Scott_theorem" title="O'Nan–Scott theorem">O'Nan–Scott theorem</a>, a classification of finite primitive groups into various types</li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-1"><a href="#cite_ref-1">^</a> Galois' last letter: <a rel="nofollow" class="external free" href="http://www.galois.ihp.fr/ressources/vie-et-oeuvre-de-galois/lettres/lettre-testament">http://www.galois.ihp.fr/ressources/vie-et-oeuvre-de-galois/lettres/lettre-testament</a>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> Galois used a different terminology, because most of the terminology in this statement was introduced afterwards, partly for clarifying the concepts introduced by Galois.

</li>

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<ul><li><a href="Colva_Roney-Dougal" title="Colva Roney-Dougal">Roney-Dougal, Colva M.</a> The primitive permutation groups of degree less than 2500, <a href="Journal_of_Algebra" title="Journal of Algebra">Journal of Algebra</a> 292 (2005), no. 1, 154–183.</li>

<li>The <a rel="nofollow" class="external text" href="http://www.gap-system.org">GAP</a> <a rel="nofollow" class="external text" href="http://www.gap-system.org/Datalib/prim.html">Data Library "Primitive Permutation Groups"</a>.</li>

<li>Carmichael, Robert D., Introduction to the Theory of Groups of Finite Order. Ginn, Boston, 1937. Reprinted by Dover Publications, New York, 1956.</li>

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</style><cite id="CITEREFWeisstein" class="citation web cs1">Todd Rowland. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/PrimitiveGroupAction.html">"Primitive Group Action"</a>. <a href="MathWorld" title="MathWorld">MathWorld</a>.</cite></li></ul></div><div><div class="zim-footer">

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