Question

8. Let G = {[1], [5], [7], [11]}, where [a] = {x ? ? : x ? a (mod 12)}. (a) Draw the Cayley table for (G, ·) where · is the operation of multiplication modulo 12. (b) Use your Cayley table to prove that (G, ·) is a group. You may assume that the operation · is associative. (c) From class we know that (??, +) and (?? × ??, +) are two non-isomorphic groups that each have four elements. Which one of these groups is isomorphic to (G, ·)? Explain your answer briefly.

          8. Let G = {[1], [5], [7], [11]}, where [a] = {x ? ? : x ? a (mod 12)}.
(a) Draw the Cayley table for (G, ·) where · is the operation of multiplication modulo 12.
(b) Use your Cayley table to prove that (G, ·) is a group. You may assume that the operation · is associative.
(c) From class we know that (??, +) and (?? × ??, +) are two non-isomorphic groups that each have four elements. Which one of these groups is isomorphic to (G, ·)? Explain your answer briefly.

8. Let G = [1], [5], [7], [11], where [a] = x ? ? : x ? a (mod 12).
(a) Draw the Cayley table for (G, ·) where · is the operation of multiplication modulo 12.
(b) Use your Cayley table to prove that (G, ·) is a group. You may assume that the operation · is associative.
(c) From class we know that (??, +) and (?? × ??, +) are two non-isomorphic groups that each have four elements. Which one of these groups is isomorphic to (G, ·)? Explain your answer briefly.

Added by Russell L.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Sri K

01/17/2023

Step 1

(a) The Cayley table for (G, *) where * is the operation of multiplication modulo 12 is as follows: | | [1] | [5] | [7] | [11] | |---|-----|-----|-----|------| | [1] | [1] | [5] | [7] | [11] | | [5] | [5] | [1] | [11] | [7] | | [7] | [7] | [11] | [1] | [5] | | Show more…

Show all steps

Thanks for your feedback!

8. Let G = {[1], [5], [7], [11]}, where [a] = {x ∈ ℤ : x ≡ a (mod 12)}. (a) Draw the Cayley table for (G, ·) where · is the operation of multiplication modulo 12. (b) Use your Cayley table to prove that (G, ·) is a group. You may assume that the operation · is associative. (c) From class we know that (ℤ₄, +) and (ℤ₂ × ℤ₂, +) are two non-isomorphic groups that each have four elements. Which one of these groups is isomorphic to (G, ·)? Explain your answer briefly.