Join operator: Difference between revisions

Latest revision as of 19:57, 21 September 2008

This is a binary subgroup property operator, viz an operator that takes as input two subgroup properties, and outputs one subgroup property

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Definition

Definition with symbols

Given two subgroup properties $p$ and $q$ , the join of $p$ and $q$ , denoted as $\langle p,q\rangle$ , is defined as follows: a subgroup $H$ satisfies property $\langle p,q\rangle$ in $G$ if there exist subgroups $K_{1},K_{2}$ of $H$ such that $K_{1}$ satisfies $p$ in $G$ , $K_{2}$ satisfies $q$ in $G$ , and $H=<K_{1},K_{2}>$ .

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	===Definition with symbols===		===Definition with symbols===

	Given two subgroup properties <math>p</math> and <math>q</math>, the join of <math>p</math> and <math>q</math>, denoted as <math>p \~~cup q~~</math>, is defined as follows: a subgroup <math>H</math> satisfies property <math>p \~~cup q~~</math> in <math>G</math> if there exist subgroups <math>K_1,K_2</math> of <math>H</math> such that <math>K_1</math> satisfies <math>p</math> in <math>G</math>, <math>K_2</math> satisfies <math>q</math> in <math>G</math>, and <math>H = <K_1,K_2></math>.		Given two subgroup properties <math>p</math> and <math>q</math>, the join of <math>p</math> and <math>q</math>, denoted as <math>\langle p,q \rangle</math>, is defined as follows: a subgroup <math>H</math> satisfies property <math>\langle p, q \rangle</math> in <math>G</math> if there exist subgroups <math>K_1,K_2</math> of <math>H</math> such that <math>K_1</math> satisfies <math>p</math> in <math>G</math>, <math>K_2</math> satisfies <math>q</math> in <math>G</math>, and <math>H = <K_1,K_2></math>.