Varietal group property - Revision history

R-a-jones at 22:05, 12 January 2024

2024-01-12T22:05:26Z

← Older revision		Revision as of 22:05, 12 January 2024
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	==Definition==		==Definition==

	A [[group property]] <math>\alpha</math> is termed '''~~Varietal~~''' if it satisfies the following three conditions:		A [[group property]] <math>\alpha</math> is termed '''varietal''' if it satisfies the following three conditions:

	# It is a [[subgroup-closed group property]], i.e., whenever <math>G</matH> is a group satisfying <math>\alpha</math> and <math>H</math> is a subgroup of <math>G</math>, <math>H</math> also satisfies <math>\alpha</math>.		# It is a [[subgroup-closed group property]], i.e., whenever <math>G</matH> is a group satisfying <math>\alpha</math> and <math>H</math> is a subgroup of <math>G</math>, <math>H</math> also satisfies <math>\alpha</math>.

R-a-jones: /* Relation with other metaproperties */

2024-01-12T22:05:11Z

Relation with other metaproperties

← Older revision		Revision as of 22:05, 12 January 2024
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	\| [[Stronger than::quotient-closed group property]] \|\| closed under taking [[quotient group]]s \|\| \|\| \|\| {{intermediate notions short\|quotient-closed group property\|varietal group property}}		\| [[Stronger than::quotient-closed group property]] \|\| closed under taking [[quotient group]]s \|\| \|\| \|\| {{intermediate notions short\|quotient-closed group property\|varietal group property}}
	\|-		\|-
	\| [[Stronger than::~~finite~~ direct product-closed group property]] \|\| closed under taking ~~finite~~ [[direct product]]s \|\| \|\| \|\| {{intermediate notions short\|~~finite~~ direct product-closed group property\|varietal group property}}		\| [[Stronger than::direct product-closed group property]] \|\| closed under taking [[direct product]]s \|\| \|\| \|\| {{intermediate notions short\|direct product-closed group property\|varietal group property}}
	\|}		\|}

R-a-jones: /* Relation with other metaproperties */

2024-01-12T22:04:03Z

Relation with other metaproperties

← Older revision		Revision as of 22:04, 12 January 2024
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	==Relation with other metaproperties==		==Relation with other metaproperties==

	~~===Stronger metaproperties===~~

	~~{\| class="sortable" border="1"~~
	~~! Metaproperty !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions~~

	\|}

	===Weaker metaproperties===		===Weaker metaproperties===

R-a-jones: Created page with "{{group metaproperty}} ==Definition== A group property $\alpha$ is termed '''Varietal''' if it satisfies the following three conditions: # It is a subgroup..."

2024-01-12T22:03:52Z

Created page with "{{group metaproperty}} ==Definition== A group property <math>\alpha</math> is termed '''Varietal''' if it satisfies the following three conditions: # It is a subgroup..."

New page

{{group metaproperty}}

==Definition==

A [[group property]] <math>\alpha</math> is termed '''Varietal''' if it satisfies the following three conditions:

# It is a [[subgroup-closed group property]], i.e., whenever <math>G</matH> is a group satisfying <math>\alpha</math> and <math>H</math> is a subgroup of <math>G</math>, <math>H</math> also satisfies <math>\alpha</math>.
# It is a [[quotient-closed group property]], i.e., whenever <math>G</math> is a group satisfying <math>\alpha</math> and <math>H</math> is a [[normal subgroup]] of <math>G</math>, the [[quotient group]] <math>G/H</math> also satisfies <math>\alpha</math>.
# It is a [[direct product-closed group property]], i.e., whenever <math>G_1,G_2,\dots</math> are groups all of which satisfy <math>\alpha</math>, the [[external direct product]] <math>G_1 \times G_2 \times \dots \times G_n</math> for any <math>n</math>, and the infinite direct product <math>G_1 \times G_2 \times \dots </math> also satisfy <math>\alpha</math>.

==Relation with other metaproperties==

===Stronger metaproperties===

{| class="sortable" border="1"
! Metaproperty !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions

|}

===Weaker metaproperties===

{| class="sortable" border="1"
! Metaproperty !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Stronger than::pseudovarietal group property]] || subgroup-closed, quotient-closed, and closed under ''finite'' direct products || || || {{intermediate notions short|pseudovarietal group property|varietal group property}}
|-
| [[Stronger than::subgroup-closed group property]] || closed under taking [[subgroup]]s || || || {{intermediate notions short|subgroup-closed group property|varietal group property}}
|-
| [[Stronger than::quotient-closed group property]] || closed under taking [[quotient group]]s || || || {{intermediate notions short|quotient-closed group property|varietal group property}}
|-
| [[Stronger than::finite direct product-closed group property]] || closed under taking finite [[direct product]]s || || || {{intermediate notions short|finite direct product-closed group property|varietal group property}}
|}

Varietal group property - Revision history

R-a-jones at 22:05, 12 January 2024

R-a-jones: /* Relation with other metaproperties */

R-a-jones: /* Relation with other metaproperties */

R-a-jones: Created page with "{{group metaproperty}} ==Definition== A group property \alpha is termed '''Varietal''' if it satisfies the following three conditions: # It is a subgroup..."

R-a-jones: Created page with "{{group metaproperty}} ==Definition== A group property $\alpha$ is termed '''Varietal''' if it satisfies the following three conditions: # It is a subgroup..."