Question

Let (G, :) such that G = {1, - 1, i, - i} is the set of four complex numbers and · is the operation of complex multiplication. Show that G is a group and find its unity and inverses. Furthermore, let (Z4, Ⓡ ) be the additive group as per Example 1.4(iv). Define the function [G, Z., f] as f(1) = [0]4; $( - 1) = [1]4, f (i) = [2]4, f ( – i) = [3]4. Show that f is an isomor- phism.

Answer 1

with G= {1,-1, i, - i} with respect to complex multiplication. 6 G Z 0 elements ab € G ab E G. See this table two (22) for any --- CAYLEY TABLE - į i 1 -i i ili -i-1 1 COMPOSITION TABLE (iv) Inverse of any (w) from the above table 1 is the unity element exist so from the table. (1) - 1 , (-15 = -1, (i)=-i # (-i) Ei Now f: G Z4 , & is a bijection defined as f(1) =[],, f1) = [114,4 (0) = 1214,4+1) [3], G& 24 has some no. of elements we show fis homomorphism, then f becomes an is omorphism, wohe So, If

But if we show that f is not a homomorphism then of is not isomorphism See f(-1,-1) = f(1) [214 if (-1) f(-i) = [1] + [3] [4] = [o] so, f. (-1, "i) & f(-1) + f(-i) Hince of is not an isomorphism question (last part) is wrong.
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