Elliptic subgroup: Difference between revisions

Revision as of 19:32, 9 March 2009

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]
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This subgroup property is a finitarily tautological subgroup property: when the ambient group is a finite group, the property is satisfied.
View other such subgroup properties

Definition

Symbol-free definition

A subgroup of a group is termed elliptic if it forms an elliptic pair of subgroups with every subgroup of the group.

Definition with symbols

A subgroup $H$ of a group $G$ is termed elliptic if for any subgroup $K$ of $G$ , $(H, K)$ form an elliptic pair of subgroups. In other words, there exists an $n$ such that:

$⟨ H, K ⟩ = (H K)^{n} : = H K H K H K \dots H K$

where each is written $n$ times.

Relation with other properties

Stronger properties

@@ Line 12: / Line 12: @@
 A subgroup <math>H</math> of a group <math>G</math> is termed '''elliptic''' if for any subgroup <math>K</math> of <math>G</math>, <math>(H,K)</math> form an [[elliptic pair of subgroups]]. In other words, there exists an <math>n</math> such that:
-<math><H,K> = (HK)^n := HKHKHK \ldots HK</math>
+<math>\langle H,K \rangle = (HK)^n := HKHKHK \ldots HK</math>
 where each is written <math>n</math> times.