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# Problems in Group Theory

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Let G_1 and G_2 be groups and ?:G_1?G_2 a map. Which of the following is a group homomorphism? Explain your answers. If ? is a homomorphism, describe the kernel and the image of ?.

a) G_1=C_4=?a|a^4=e?,G_2=Z_2 (the integers modulo 2 with the operation +), ?:a^i?i (mod 2).

b) G_1=G_2=Z_5 (the integers modulo 5 with the operation +), ?:n?an(mod5) where a?Z_5{0}={1,2,3,4} is fixed.

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Let (please see the attached file) and (please see the attached file) be groups and (please see the attached file) a map. Which of the following is a group homomorphism? Explain your answers. If (please see the attached file) is a homomorphism, describe the kernel and the image of.
a) (the integers modulo 2 with the operation), (mod 2).
This is a group homomorphism since The kernel of is the subgroup and the image of is (please see the attached file), i.e. (please see the attached file) is an epimorphism (onto).
b) (the integers modulo 5 with the operation), (mod5) where (please see the attached file) is fixed.
This is also a group homomorphism since 5 is prime, whence multiplaiction by every integer modulo 5 induces a permutation of {1,2,3,4}. The kernel of is {0} and the image of is , i.e. is a group isomorphism.

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