Question

Let K = gp([4]12) be a subset of the group Z12. (a) Give the members of gp([4]12). (b) Show that K is a normal subgroup of Z12, i.e. K ⊴ Z12. [Hint: A consequence of Lagrange's Theorem may be useful] (c) Give the operation table for Z12/K. (d) Show that the mapping θ: Z12 → Z4 defined by θ([a]12) = [a]4 is a (group) homomorphism. (Assume the mapping is well-defined) (e) Give ker θ - the kernel of θ. (f) Give an isomorphism from Z12/ker θ and Z4, by giving the image of each element of Z12/ker θ. [Hint: The First Isomorphism Theorem may be useful.]

          Let K = gp([4]12) be a subset of the group Z12.
(a) Give the members of gp([4]12).
(b) Show that K is a normal subgroup of Z12, i.e. K ⊴ Z12.
[Hint: A consequence of Lagrange's Theorem may be useful]
(c) Give the operation table for Z12/K.
(d) Show that the mapping θ: Z12 → Z4 defined by θ([a]12) = [a]4 is a (group) homomorphism. (Assume the mapping is well-defined)
(e) Give ker θ - the kernel of θ. 
(f) Give an isomorphism from Z12/ker θ and Z4, by giving the image of each element of Z12/ker θ. [Hint: The First Isomorphism Theorem may be useful.]

Added by Miriam A.

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