Difference between revisions of "Aufgaben:Problem 3"
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(Created page with "Let G be a finite group. For a given g \in G we consider the map L_g : G \rightarrow G, g \mapsto gg'. a) Prove that L : g \mapsto L_g defines a map G \rightarrow SymG where S...") |
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| − | + | Let G be a finite group. For a given g \in G we consider the map L_g : G \rightarrow G, g \mapsto gg'. | |
| − | + | a) Prove that L : g \mapsto L_g defines a map G \rightarrow SymG where SymG denotes the set of | |
| − | + | all invertible maps from G to G. | |
| − | + | b) Prove that the map L is injective. | |
| + | c) Prove that composing maps in SymG defines a group structure on SymG. | ||
| + | d) Prove that the map L is a homomorphism of groups. | ||
| + | e) Conclude that every finite group G can be considered a subgroup of SymG. | ||
Revision as of 11:03, 8 June 2015
Let G be a finite group. For a given g \in G we consider the map L_g : G \rightarrow G, g \mapsto gg'. a) Prove that L : g \mapsto L_g defines a map G \rightarrow SymG where SymG denotes the set of all invertible maps from G to G. b) Prove that the map L is injective. c) Prove that composing maps in SymG defines a group structure on SymG. d) Prove that the map L is a homomorphism of groups. e) Conclude that every finite group G can be considered a subgroup of SymG.
