document.write( "Question 1186053: This a Abstract Algebra class\r
\n" ); document.write( "\n" ); document.write( "Let G=⟨a⟩ be a cyclic group of order10. Define φ:Z→G by φ(n)=a^n for all n∈Z. Assume that φ is a homomorphism. Find kerφ. \n" ); document.write( "

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\n" ); document.write( "This a Abstract Algebra class
\n" ); document.write( "Let G=⟨a⟩ be a cyclic group of order10. Define φ:Z→G by φ(n)=a^n for all n∈Z.
\n" ); document.write( "Assume that φ is a homomorphism. Find kerφ.
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document.write( "The group G is presented as  G = (a) in the post.\r\n" );
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document.write( "Also, from the context, G is a multiplicative group, regarding its operation.\r\n" );
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document.write( "It means that element \"a\" of the group G is the generator of the cyclic group G.\r\n" );
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document.write( "In turn, it means that   = 1  (the unit element of the group G),  \r\n" );
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document.write( "while  the elements    at k = 1, 2, 3, . . . , 9 are all different and not equal to the unit element of the group.\r\n" );
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document.write( "Hence, kerφ is the set of all integers in Z that are multiples of 10:   { . . . , -20, -10, 0, 10, 20, . . . }.    ANSWER\r\n" );
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\n" ); document.write( "\n" ); document.write( "We should not \"assume\" that φ , defined in this way, is a homomorphism.\r
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\n" ); document.write( "\n" ); document.write( "It REALLY IS the homomorphism.\r
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