Vipul: /* Stronger properties */

2020-06-01T14:25:53Z

Stronger properties

← Older revision		Revision as of 14:25, 1 June 2020
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	! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
	\|-		\|-
	\| [[Weaker than::~~Elementarily~~ characteristic subgroup]] \|\| no other elementarily equivalent subgroup \|\| \|\| \|\|		\| [[Weaker than::elementarily characteristic subgroup]] \|\| no other elementarily equivalent subgroup \|\| \|\| \|\|
	\|-		\|-
	\| [[Weaker than::~~Monadic~~ second-order purely definable subgroup]] \|\| can be defined using the pure theory of the group in monadic second-order language \|\| \|\| \|\|		\| [[Weaker than::monadic second-order purely definable subgroup]] \|\| can be defined using the pure theory of the group in monadic second-order language \|\| \|\| \|\|
	\|-		\|-
	\| [[Weaker than::~~Purely~~ definable subgroup]] \|\| can be defined purely using the first-order pure theory of the group \|\| \|\| \|\|		\| [[Weaker than::purely definable subgroup]] \|\| can be defined purely using the first-order pure theory of the group \|\| \|\| \|\|
	\|}		\|}

Vipul: Created page with '{{wikilocal}} {{subgroup property}} ==Definition== A subgroup $H$ of a group $G$ is termed a '''monadic second-order characteristic subgroup''' if…'

2010-01-05T22:24:19Z

Created page with '{{wikilocal}} {{subgroup property}} ==Definition== A subgroup <math>H</math> of a group <math>G</math> is termed a '''monadic second-order characteristic subgroup''' if…'

New page

{{wikilocal}}
{{subgroup property}}

==Definition==

A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed a '''monadic second-order characteristic subgroup''' if there is no other subgroup <math>K</math> of <math>G</math> such that the monadic second-order theories of the group-subgroup pairs <math>(G,H)</math> and <math>(G,K)</math> coincide. In other words, <math>H</math> can be distinguished from any other subgroup of <math>G</math> using monadic second-order logic in the pure theory of the group <math>G</math>.

==Relation with other properties==

===Stronger properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Weaker than::Elementarily characteristic subgroup]] || no other elementarily equivalent subgroup || || ||
|-
| [[Weaker than::Monadic second-order purely definable subgroup]] || can be defined using the pure theory of the group in monadic second-order language || || ||
|-
| [[Weaker than::Purely definable subgroup]] || can be defined purely using the first-order pure theory of the group || || ||
|}

===Weaker properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Stronger than::Second-order characteristic subgroup]] || || || ||
|-
| [[Stronger than::Characteristic subgroup]] || || || ||
|}

Monadic second-order characteristic subgroup - Revision history

Vipul: /* Stronger properties */

Vipul: Created page with '{{wikilocal}} {{subgroup property}} ==Definition== A subgroup H of a group G is termed a '''monadic second-order characteristic subgroup''' if…'

Vipul: Created page with '{{wikilocal}} {{subgroup property}} ==Definition== A subgroup $H$ of a group $G$ is termed a '''monadic second-order characteristic subgroup''' if…'