Bestimme die Nebenklassen zu den folgenden Untergruppen von kommutativen Gruppen. a) ( Z , 0 , + ) ⊆ ( R , 0 , + ) {\displaystyle {}(\mathbb {Z} ,0,+)\subseteq (\mathbb {R} ,0,+)} . b) ( Q , 0 , + ) ⊆ ( R , 0 , + ) {\displaystyle {}(\mathbb {Q} ,0,+)\subseteq (\mathbb {R} ,0,+)} . c) ( R , 0 , + ) ⊆ ( C , 0 , + ) {\displaystyle {}(\mathbb {R} ,0,+)\subseteq ({\mathbb {C} },0,+)} . d) ( Z d , 0 , + ) ⊆ ( Z , 0 , + ) {\displaystyle {}(\mathbb {Z} d,0,+)\subseteq (\mathbb {Z} ,0,+)} (zu d ∈ N {\displaystyle {}d\in \mathbb {N} } ). e) ( { z ∈ C ∣ | z | = 1 } , 1 , ⋅ ) ⊆ ( C ∖ { 0 } , 1 , ⋅ ) {\displaystyle {}({\left\{z\in {\mathbb {C} }\mid \vert {z}\vert =1\right\}},1,\cdot )\subseteq ({\mathbb {C} }\setminus \{0\},1,\cdot )} . f) ( { z ∈ C ∣ z n = 1 } , 1 , ⋅ ) ⊆ ( { z ∈ C ∣ | z | = 1 } , 1 , ⋅ ) {\displaystyle {}({\left\{z\in {\mathbb {C} }\mid z^{n}=1\right\}},1,\cdot )\subseteq ({\left\{z\in {\mathbb {C} }\mid \vert {z}\vert =1\right\}},1,\cdot )} (zu n ∈ N {\displaystyle {}n\in \mathbb {N} } ).
Wann bestehen die Nebenklassen aus endlich vielen Elementen, wann ist der Index endlich?
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