Iterated agemo subgroup of p-group: Difference between revisions

Latest revision as of 22:39, 8 August 2012

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties [SHOW MORE]
|Get subgroup property lookup help |Get exploration suggestions
VIEW RELATED: Subgroup property implications | Subgroup property non-implications | Subgroup metaproperty satisfactions | Subgroup metaproperty dissatisfactions | Subgroup property satisfactions |Subgroup property dissatisfactions
VIEW FACTS THAT:use, or start with, a subgroup satisfying the property|prove, or show that, a subgroup satisfies the property

Definition

Suppose $p$ is a prime number and $G$ is a p-group. A subgroup $H$ of $G$ is an iterated agemo subgroup of $G$ if there exists a sequence $a_{1}, a_{2}, \dots, a_{r}$ such that:

$H : = ℧^{a_{1}} (℧^{a_{2}} (℧^{a_{3}} (\dots (℧^{a_{r}} (G))) \dots)$ .

Relation with other properties

Stronger properties

property	quick description	proof of implication	proof of strictness (reverse implication failure)	intermediate notions
agemo subgroup of p-group			iterated agemo subgroup not implies agemo subgroup

Weaker properties

property	quick description	proof of implication	proof of strictness (reverse implication failure)	intermediate notions
verbal subgroup of p-group		iterated agemo implies verbal	verbal not implies iterated agemo

@@ Line 3: / Line 3: @@
 ==Definition==
-Suppose <math>p</math> is a [[prime number]] and <math>G</math> is a finite <math>p</math>-group, i.e., a [[group of prime power order]]. A subgroup <math>H</math> of <math>G</math> is an '''iterated agemo subgroup''' of <math>G</math> if there exists a sequence <math>a_1,a_2,\dots,a_r</math> such that:
+Suppose <math>p</math> is a [[prime number]] and <math>G</math> is a [[p-group]]. A subgroup <math>H</math> of <math>G</math> is an '''iterated agemo subgroup''' of <math>G</math> if there exists a sequence <math>a_1,a_2,\dots,a_r</math> such that:
 <math>H := \mho^{a_1}(\mho^{a_2}(\mho^{a_3}(\dots(\mho^{a_r}(G)))\dots)</math>.
@@ Line 11: / Line 11: @@
 ===Stronger properties===
-{| class="wikitable" border="1"
+{| class="sortable" border="1"
 ! property !! quick description !! proof of implication !! proof of strictness (reverse implication failure) !! intermediate notions
 |-
-| [[Weaker than::agemo subgroup of group of prime power order]] || || || [[iterated agemo subgroup not implies agemo subgroup]] ||
+| [[Weaker than::agemo subgroup of p-group]] || || || [[iterated agemo subgroup not implies agemo subgroup]] ||
 |}
 ===Weaker properties===
-{| class="wikitable" border="1"
+{| class="sortable" border="1"
 ! property !! quick description !! proof of implication !! proof of strictness (reverse implication failure) !! intermediate notions
 |-
-| [[Stronger than::Verbal subgroup of group of prime power order]] || || [[iterated agemo implies verbal]] || [[verbal not implies iterated agemo]] || {{intermediate notions short|verbal subgroup of group of prime power order|iterated agemo subgroup of group of prime power order}}
+| [[Stronger than::verbal subgroup of p-group]] || || [[iterated agemo implies verbal]] || [[verbal not implies iterated agemo]] ||
 |}