Difference between revisions of "Aufgaben:Problem 3"
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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 13:31, 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 \(SymG\).
