Amalgamated free product: Difference between revisions

Latest revision as of 00:46, 20 December 2010

This article describes a product notion for groups. See other related product notions for groups.

Definition

Definition with strict common subgroup

Let $G_{1}$ and $G_{2}$ be two groups, and let $H$ be a group with an injective homomorphism to both $G_{1}$ and $G_{2}$ . Then the amalgamated free product of $G_{1}$ and $G_{2}$ via $H$ is defined as the quotient of the free product of $G_{1}$ and $G_{2}$ , by the relation of the $H$ in $G_{1}$ being the same as the $H$ in $G_{2}$ .

Definition of weaker notion (pushout version)

PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

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 ==Definition==
+===Definition with strict common subgroup===
 Let <math>G_1</math> and <math>G_2</math> be two groups, and let <math>H</math> be a group with an injective homomorphism to both <math>G_1</math> and <math>G_2</math>. Then the amalgamated free product of <math>G_1</math> and <math>G_2</math> via <math>H</math> is defined as the quotient of the free product of <math>G_1</math> and <math>G_2</math>, by the relation of the <math>H</math> in <math>G_1</math> being the same as the <math>H</math> in <math>G_2</math>.
+===Definition of weaker notion (pushout version)===
+{{fillin}}