2. Deduce from 1 that V x 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)]¬1? What is its order in V x Z2? Justify your answers.
2. Deduce from 1 that V x 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)]¬1? What is its order in V x Z2? Justify your answers.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
answer number 2
![1. Assume (X, o) and (Y, •) are groups. Let X x Y = {(r, y)|1 € X, y € Y} and define the operation
on X x Y as
(F1, 4ı) * (x2, Y2) = (#1 0 r2, Y1 • Y2)
%3D
for (21, y1), (x2, Y2) e X × Y. Show that (X x Y, *) is a group.
2. Deduce from 1 that V x Z2 is a group where V = {e, a, b, c} is the Klein-4 group.
%3D
(a) Give its Cayley Table.
(b) What is [(a, 1) * (b, 1)]1? what is its order in V x Z2? Justify your answers.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F43f8b2c2-a9ca-4903-9d0d-cd293e490b2f%2F08edcfaf-d1a1-4cb7-af73-3d65a462a645%2Flt9g8ie_processed.png&w=3840&q=75)
Transcribed Image Text:1. Assume (X, o) and (Y, •) are groups. Let X x Y = {(r, y)|1 € X, y € Y} and define the operation
on X x Y as
(F1, 4ı) * (x2, Y2) = (#1 0 r2, Y1 • Y2)
%3D
for (21, y1), (x2, Y2) e X × Y. Show that (X x Y, *) is a group.
2. Deduce from 1 that V x Z2 is a group where V = {e, a, b, c} is the Klein-4 group.
%3D
(a) Give its Cayley Table.
(b) What is [(a, 1) * (b, 1)]1? what is its order in V x Z2? Justify your answers.
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