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Isomorphism Theorem

Proposition (First Isomorphism Theorem) Let isomorphism theorem _gr_1.gif] and isomorphism theorem _gr_2.gif] be groups, and let isomorphism theorem _gr_3.gif] be a homomorphism from isomorphism theorem _gr_4.gif] onto isomorphism theorem _gr_5.gif] with isomorphism theorem _gr_6.gif] Then the mapping isomorphism theorem _gr_7.gif] defined by isomorphism theorem _gr_8.gif] for each isomorphism theorem _gr_9.gif] is an isomorphism of isomorphism theorem _gr_10.gif] onto isomorphism theorem _gr_11.gif]

    Proof. To show that isomorphism theorem _gr_12.gif] is well-defined let isomorphism theorem _gr_13.gif] Then isomorphism theorem _gr_14.gif] for some isomorphism theorem _gr_15.gif] so isomorphism theorem _gr_16.gif] isomorphism theorem _gr_17.gif] isomorphism theorem _gr_18.gif] isomorphism theorem _gr_19.gif] isomorphism theorem _gr_20.gif]Thus, isomorphism theorem _gr_21.gif] implies isomorphism theorem _gr_22.gif] and so isomorphism theorem _gr_23.gif] is a mapping. Since isomorphism theorem _gr_24.gif] isomorphism theorem _gr_25.gif] isomorphism theorem _gr_26.gif] isomorphism theorem _gr_27.gif] isomorphism theorem _gr_28.gif] for all isomorphism theorem _gr_29.gif] isomorphism theorem _gr_30.gif] preserves the group operations. Clearly, isomorphism theorem _gr_31.gif] is onto because isomorphism theorem _gr_32.gif] is onto isomorphism theorem _gr_33.gif] To show that isomorphism theorem _gr_34.gif] is one-to-one we will show that isomorphism theorem _gr_35.gif] If isomorphism theorem _gr_36.gif] then isomorphism theorem _gr_37.gif] isomorphism theorem _gr_38.gif] isomorphism theorem _gr_39.gif] and so isomorphism theorem _gr_40.gif] Therefore, isomorphism theorem _gr_41.gif] as desired. isomorphism theorem _gr_42.gif]

    The natural homomorphism isomorphism theorem _gr_43.gif] shows that each quotient group is the homomorphic image of isomorphism theorem _gr_44.gif] and the First Isomorphism Theorem states that the converse is also true; that is, each homomorphic image of isomorphism theorem _gr_45.gif] is a quotient group. Indeed, if a group homomorphism is not onto then you can replace isomorphism theorem _gr_46.gif] by isomorphism theorem _gr_47.gif] and thus we have, isomorphism theorem _gr_48.gif] in the following commutative diagram.

isomorphism theorem _gr_49.gif]

Example (First Isomorphism Theorem) For isomorphism theorem _gr_50.gif] let isomorphism theorem _gr_51.gif] and isomorphism theorem _gr_52.gif] denote the congruence classes determined by isomorphism theorem _gr_53.gif] in isomorphism theorem _gr_54.gif] and isomorphism theorem _gr_55.gif] respectively. Define isomorphism theorem _gr_56.gif] by isomorphism theorem _gr_57.gif] Then isomorphism theorem _gr_58.gif] is well-defined because if isomorphism theorem _gr_59.gif] then isomorphism theorem _gr_60.gif] and therefore, isomorphism theorem _gr_61.gif] and isomorphism theorem _gr_62.gif] Since isomorphism theorem _gr_63.gif] isomorphism theorem _gr_64.gif] isomorphism theorem _gr_65.gif] isomorphism theorem _gr_66.gif] isomorphism theorem _gr_67.gif] isomorphism theorem _gr_68.gif] is a homomorphism. Clearly, isomorphism theorem _gr_69.gif] is onto and since isomorphism theorem _gr_70.gif] isomorphism theorem _gr_71.gif] isomorphism theorem _gr_72.gif] the First Isomorphism Theorem yields isomorphism theorem _gr_73.gif] isomorphism theorem _gr_74.gif]

Example (First Isomorphism Theorem) Consider the multiplicative group isomorphism theorem _gr_75.gif] of nonzero complex numbers. Let isomorphism theorem _gr_76.gif] be the set of all complex numbers of absolute value isomorphism theorem _gr_77.gif] Then isomorphism theorem _gr_78.gif] is a normal subgroup of isomorphism theorem _gr_79.gif] and the quotient group of isomorphism theorem _gr_80.gif] is isomorphic to the multiplicative group isomorphism theorem _gr_81.gif] of all positive real numbers. To see this, use the First Homomorpism Theorem applied to the onto homomorphism isomorphism theorem _gr_82.gif] defined by isomorphism theorem _gr_83.gif] with isomorphism theorem _gr_84.gif] isomorphism theorem _gr_85.gif]

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Isomorphism Theorem
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