Exterior product of groups: Difference between revisions

Revision as of 14:14, 13 June 2012

Definition

Suppose $G, H$ are (possibly equal, possibly distinct) normal subgroups of some group $Q$ . (Note that it in fact suffices to assume that they normalize each other, but there is no loss of generality in assuming they are both normal).

Define a compatible pair of actions of $G$ and $H$ on each other by each acting on the other as conjugation in $Q$ . The exterior product $G \land H$ is defined as the quotient group of the tensor product of groups $G \otimes H$ for this compatible pair of actions by the normal subgroup generated by all elements of the form $x \otimes x, x \in G \cap H$ .

The image of the symbol $g \otimes h$ in the quotient is denoted $g \land h$ .

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 Suppose <math>G,H</math> are (possibly equal, possibly distinct) [[normal subgroup]]s of some group <math>Q</math>. (Note that it in fact suffices to assume that they normalize each other, but there is no loss of generality in assuming they are both normal).
-Define a [[compatible pair of actions]] of <math>G</math> and <math>H</math> on each other by each actin on the other as conjugation in <math>Q</math>. The '''exterior product''' <math>G \wedge H</math> is defined as the [[quotient group]] of the [[tensor product of groups]] <math>G \otimes H</math> for this compatible pair of actions by the [[normal subgroup generated by a subset|normal subgroup generated]] by all elements of the form <math>x \otimes x, x \in G \cap H</math>.
+Define a [[compatible pair of actions]] of <math>G</math> and <math>H</math> on each other by each acting on the other as conjugation in <math>Q</math>. The '''exterior product''' <math>G \wedge H</math> is defined as the [[quotient group]] of the [[tensor product of groups]] <math>G \otimes H</math> for this compatible pair of actions by the [[normal subgroup generated by a subset|normal subgroup generated]] by all elements of the form <math>x \otimes x, x \in G \cap H</math>.
 The image of the symbol <math>g \otimes h</math> in the quotient is denoted <math>g \wedge h</math>.