1. Assume (X, o) and (Y, on X x Y as are groups. Let X × Y = {(x, y) |x € X, y € Y} and define the operation * (x1, Y1) * (x2, Y2) = (x1 0 x2, Y1 • Y2) for (x1, y1), (x2,Y2) E X × Y. Show that (X x Y,...

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Extracted text: 1. Assume (X, o) and (Y, on X x Y as are groups. Let X × Y = {(x, y) |x € X, y € Y} and define the operation * (x1, Y1) * (x2, Y2) = (x1 0 x2, Y1 • Y2) for (x1, y1), (x2,Y2) E X × Y. Show that (X x Y, *) is a group.


Extracted text: 2. Deduce from 1 that V × Z2 is a group where V = {e, a, b, c} is the Klein-4 group. (a) Give its Cayley Table. (b) What is [(a, 1) * (b, 1)]¯!? What is its order in V × Z2? Justify your answers.