MAT 2143, Winter 2022 Fourth Assignment
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MAT 2143, Winter 2022
Fourth Assignment
1. Consider the map φ : z x z → z2 x z4 given by
φ(x, y) := (x mod 2, (x + 2y) mod 4).
(a) [2 points] Prove that φ is a group homomorphism.
(b) [3 points] Determine ker(φ).
Hint: Try a case-by-case analysis: given (x, y) e z x z, consider several cases based on the remainders of x and y modulo 2 and 4.
(c) [2 points] Determine im(φ).
2. [4 points] Let G be a finite group and let H and K be normal subgroups of G such that H n K = {e} and G = HK. prove that the map
ψ : G → G/H x G/K , ψ(g) := (gH, gK)
is an isomorphism of groups.
2022-04-06