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- 21:24, 16 August 2026 Expected square of number of fixed points of permutation equals two (hist | edit) [2,614 bytes] Vipul (talk | contribs) (Created page with "==Statement== Suppose <math>n</math> is a natural number. Consider the uniform distribution on the symmetric group of degree <math>n</math>, i.e., the symmetric group on a set of size <math>n</math>. For <math>n \ge 2</math>, the expected value of the square of the number of fixed points for a permutation picked according to the uniform probability distribution equals <math>2</math>. See also probability distribution of number of...")
- 02:09, 6 August 2026 Eventually constant-abelian-quotient implies contrasaturated (hist | edit) [3,820 bytes] Vipul (talk | contribs) (Created page with "{{group sub-IAPS property implication |stronger = eventually constant-abelian-quotient sub-IAPS |weaker = contrasaturated normal sub-IAPS }} ==Statement== Let <math>G</math> be an IAPS of groups and <math>H</math> be an eventually constant-abelian-quotient sub-IAPS, i.e., <math>H</math> is normal in <math>G</math> and there exists <math>n_0</math> such that <math>G_n / H_n</math> is isomorphic to a fixed abelian group <math>A</math> for...")
- 01:46, 6 August 2026 Constant-abelian-quotient not implies finitely displaced (hist | edit) [2,055 bytes] Vipul (talk | contribs) (Created page with "{{group sub-IAPS property non-implication |stronger = eventually constant-abelian-quotient sub-IAPS |weaker = finitely displaced sub-IAPS}} ==Statement== It is possible to have an IAPS of groups <math>G</math> and a sub-IAPS <math>H</math> such that <math>H</math> is constant-abelian-quotient in <math>G</math> but is not a finitely displaced sub-IAPS of <math>G</math>. In fact, we can take <math>G</math> to be a power AP...")
- 00:07, 6 August 2026 Power sub-APS of power APS may be finitely displaced (hist | edit) [1,568 bytes] Vipul (talk | contribs) (Created page with "==Statement== It is possible to have a nontrivial group <math>G</math> (in fact, <math>G</math> can be chosen as an abelian group) and a subgroup <math>H</math> such that the power APS of <math>H</math> is a finitely displaced sub-IAPS inside the power APS of <math>G</math>. Since power sub-APSes are saturated by construction, this gives an example of a strongly proper finitely displaced saturated s...")
- 23:21, 5 August 2026 Padding-saturated sub-IAPS (hist | edit) [2,269 bytes] Vipul (talk | contribs) (Created page with "{{group sub-IAPS property}} ==Definition== Let <math>(G,\Phi)</math> be an IAPS of groups and <math>H</math> be a sub-IAPS. We say that <math>H</math is '''padding-saturated''' in <math>G</math> if both these conditions hold for all positive integers <math>m</math> and <math>n</math>: * The intersection of <math>H_{m + n}</math> with the image of the restriction of the concatenation map <math>\Phi_{m,n}: G_m \times G_n \to G_{m + n}</math> to...")
- 20:24, 5 August 2026 Finitely displaced normal sub-IAPS (hist | edit) [2,544 bytes] Vipul (talk | contribs) (Created page with "{{group sub-IAPS property}} ==Definition== Let <math>(G,\Phi)</math> be an IAPS of groups and <math>H</math> be a sub-IAPS. We say that <math>H</math is a '''finitely displaced normal sub-IAPS''' if <math>H</math> is both finitely displaced and normal in <math>G</math>. ==Relation with other properties== ===Stronger properties=== {| class=...")
- 19:40, 5 August 2026 Eventually constant-abelian-quotient sub-IAPS (hist | edit) [2,725 bytes] Vipul (talk | contribs) (Created page with "{{group sub-IAPS property}} ==Definition== Let <math>(G,\Phi)</math> be an IAPS of groups and <math>H</math> a sub-IAPS. We say that <math>H</math> is an '''eventually constant-abelian-quotient sub-IAPS''' if <math>H</math> is normal in <math>G</math> and the quotient APS <math>G/H</math> is eventually isomorphic to the defining ingredient::constant APS of an abelian group. =...")
- 19:33, 5 August 2026 Constant-abelian-quotient sub-IAPS (hist | edit) [2,363 bytes] Vipul (talk | contribs) (Created page with "{{group sub-IAPS property}} ==Definition== Let <math>(G,\Phi)</math> be an IAPS of groups and <math>H</math> a sub-IAPS. We say that <math>H</math> is a '''constant-abelian-quotient sub-IAPS''' if <math>H</math> is normal in <math>G</math> and the quotient APS <math>G/H</math> is the defining ingredient::constant APS of an abelian group. ==Examples== * In the GL IAPS for a unital ring, the special linear sub-IAPS is constant-abelian...")
- 19:25, 5 August 2026 Abelian-quotient sub-IAPS (hist | edit) [2,282 bytes] Vipul (talk | contribs) (Created page with "{{group sub-IAPS property}} ==Definition== Let <math>(G,\Phi)</math> be an IAPS of groups and <math>H</math> be a sub-IAPS. We say that <math>H<math> is an '''abelian-quotient sub-IAPS''' if <math>H</math> is normal and the quotient APS of groups <math>G/H</math> comprises only abelian groups. ==Relation with other properties== ===Weaker properties=== {| class="sortable" border="1" ! Property !! Meaning !! Proof of implication !! Pro...")
- 19:08, 5 August 2026 Contrasaturated normal sub-IAPS (hist | edit) [3,730 bytes] Vipul (talk | contribs) (Created page with "{{group sub-IAPS property}} ==Definition== Let <math>(G,\Phi)</math> be an IAPS of groups and <math>H</math> a sub-IAPS of <math>G</math>. We say that <math>H</math> is '''contrasaturated normal''' inside <math>G</math> if <math>H</math> is both contrasaturated and a normal. ==Examples== * In the GL IAPS, the sub-IAPS comprising special linear groups is a contrasaturated...")
- 18:38, 5 August 2026 Eventually isomorphic APSes of groups (hist | edit) [434 bytes] Vipul (talk | contribs) (Created page with "==Definition== Let <math>(G,\Phi)</math> and <math>(H,\Psi)</math> be two APSes of groups. We say that <math>G</math> and <math>H</math> are '''eventually isomorphic''' if there exists a natural number <math>n_0</math> and isomorphisms <math>G_n \to H_n</math> for all <math>n \ge n_0</math> that are compatible with the concatenation maps <math>\Phi<math> and <math>\Psi</math>.")
- 18:33, 5 August 2026 Strongly proper sub-APS (hist | edit) [266 bytes] Vipul (talk | contribs) (Created page with "{{group sub-APS property}} ==Definition== Let <math>(G,\Phi)</math> be an APS of groups. A sub-APS <math>H</math> of <math>G</math> is termed '''strongly proper''' if <math>H_m</math> is a proper subgroup of <math>G_m</math> for infinitely many <math>m</math>.")
- 18:29, 5 August 2026 Group sub-IAPS property (hist | edit) [391 bytes] Vipul (talk | contribs) (Created page with "{{wikilocal}} {{particularcases|Category: Group sub-IAPS properties}} ==Definition== A '''group sub-IAPS property''' is a map from the collection of all pairs of an IAPS of groups and a sub-IAPS to the two-element set (true, false), with the property that two equivalent (IAPS, sub-IAPS) pairs either both have the property or both do not have the property.")